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AN  INVESTIGATION 


THE  LAWS  OF  THOUGHT 


AN  INVESTIGATION 


OF 

THE  LAWS  OF  THOUGHT 


ON  WHICH  ARE  FOUNDED 

THE  MATHEMATICAL  THE0E1ES  OF  LOGIC 
AND  PBOBABILITIES. 


GEORGE  BOOLE,  LL.  D. 

PROFESSOR  OF  MATHEMATICS  IN  QUEEN’S  COLLEGE,  CORK. 


DOVER  PUBLICATIONS,  INC. 


First  American  Printing 
of  the  1854  edition  with  all  corrections 
made  within  the  text 


DOVER  PUBLICATIONS,  INC. 

1780  BROADWAY,  NEW  YORK  19,  N.  Y. 


Printed  and  bound  in  the  United  States 
of  America 


no 

37  zm 


TO 

JOHN  RY  ALL,  LL.  D. 


VICE-PRESIDENT  AND  PROFESSOR  OF  GREEK 
IN  QUEEN’S  COLLEGE,  CORK, 

THIS  WORK  IS  INSCRIBED 
IN  TESTIMONY  OF  FRIENDSHIP  AND  ESTEEM 


*,  oq 

o*  +-J  -jl*s 


PREFACE. 


f I ''HE  following  work  is  not  a republication  of  a former  trea- 
tise  by  the  Author,  entitled,  “ The  Mathematical  Analysis 
of  Logic.”  Its  earlier  portion  is  indeed  devoted  to  the  same 
object,  and  it  begins  by  establishing  the  same  system  of  funda- 
mental laws,  but  its  methods  are  more  general,  and  its  range  of 
applications  far  wider.  It  exhibits  the  results,  matured  by  some 
years  of  study  and  reflection,  of  a principle  of  investigation  re- 
lating to  the  intellectual  operations,  the  previous  exposition  of 
which  was  written  within  a few  weeks  after  its  idea  had  been 
conceived. 

That  portion  of  this  work  which  relates  to  Logic  presupposes 
in  its  reader  a knowledge  of  the  most  important  terms  of  the 
science,  as  usually  treated,  and  of  its  general  object.  On  these 
points  there  is  no  better  guide  than  Archbishop  Whately’s 
“ Elements  of  Logic,”  or  Mr.  Thomson’s  “ Outlines  of  the  Laws 
of  Thought.”  To  the  former  of  these  treatises,  the  present  re- 
vival of  attention  to  this  class  of  studies  seems  in  a great  measure 
due.  Some  acquaintance  with  the  principles  of  Algebra  is  also 
requisite,  but  it  is  not  necessary  that  this  application  should  have 
been  carried  beyond  the  solution  of  simple  equations.  For  the 
study  of  those  chapters  which  relate  to  the  theory  of  probabilities, 
a somewhat  larger  knowledge  of  Algebra  is  required,  and  espe- 
cr<o  on 


TKEFACE. 


daily  of  the  doctrine  of  Elimination,  and  of  the  solution  of  Equa- 
tions containing  more  than  one  unknown  quantity.  Preliminary 
information  upon  the  subject-matter  will  be  found  in  the  special 
treatises  on  Probabilities  in  “ Lardner’s  Cabinet  Cyclopaedia,” 
and  the  “ Library  of  Useful  Knowledge,”  the  former  of  these  by 
Professor  De  Morgan,  the  latter  by  Sir  John  Lubbock;  and  in 
an  interesting  series  of  Letters  translated  from  the  French  of 
M.  Quetelet.  Other  references  will  be  given  in  the  work.  On  a 
first  perusal  the  reader  may  omit  at  his  discretion,  Chapters  x., 
xiv.,  and  xix.,  together  with  any  of  the  applications  which  he 
may  deem  uninviting  or  irrelevant. 

In  different  parts  of  the  work,  and  especially  in  the  notes  to 
the  concluding  chapter,  will  be  found  references  to  various  writers, 
ancient  and  modern,  chiefly  designed  to  illustrate  a certain  view  of 
the  history  of  philosophy.  ' With  respect  to  these,  the  Author 
thinks  it  proper  to  add,  that  he  has  in  no  instance  given  a cita- 
tion which  he  has  not  believed  upon  careful  examination  to  be 
supported  either  by  parallel  authorities,  or  by  the  general  tenor 
of  the  work  from  which  it  was  taken.  While  he  would  gladly 
have  avoided  the  introduction  of  anything  which  might  by  pos- 
sibility be  construed  into  the  parade  of  learning,  he  felt  it  to  be 
due  both  to  his  subject  and  to  the  truth,  that  the  statements  in 
the  text  should  be  accompanied  by  the  means  of  verification. 
And  if  now,  in  bringing  to  its  close  a labour,  of  the  extent  of 
which  few  persons  will  be  able  to  judge  from  its  apparent  fruits, 
he  may  be  permitted  to  speak  for  a single  moment  of  the  feelings 
with  which  he  has  pursued,  and  with  Avhich  he  now  lays  aside, 
his  task,  he  would  say,  that  he  never  doubted  that  it  was  worthy  of 
his  best  efforts ; that  he  felt  that  whatever  of  truth  it  might  bring 
to  light  was  not  a private  or  arbitrary  thing,  not  dependent,  as  to 
its  essence,  upon  any  human  opinion.  He  was  fully  aware  that 
learned  and  able  men  maintained  opinions  upon  the  subject  of 


PREFACE. 


Logic  directly  opposed  to  the  views  upon  which  the  entire  argu- 
ment and  procedure  of  his  work  rested.  While  he  believed  those 
opinions  to  be  erroneous,  he  was  conscious  that  his  own  views 
might  insensibly  be  warped  by  an  influence  of  another  kind.  He 
felt  in  an  especial  manner  the  danger  of  that  intellectual  bias  which 
long  attention  to  a particular  aspect  of  truth  tends  to  produce. 
But  he  trusts  that  out  of  this  conflict  of  opinions  the  same  truth 
will  but  emerge  the  more  free  from  any  personal  admixture ; that 
its  different  parts  will  be  seen  in  their  just  proportion;  and  that 
none  of  them  will  eventually  be  too  highly  valued  or  too  lightly 
regarded  because  of  the  prejudices  which  may  attach  to  the 
mere  form  of  its  exposition. 

To  his  valued  friend,  the  Rev.  George  Stephens  Dickson, 
of  Lincoln,  the  Author  desires  to  record  his  obligations  for  much 
kind  assistance  in  the  revision  of  this  work,  and  for  some  impor- 
tant suggestions. 

5,  Grenville-place,  Cork, 

Nov.  30 th.  1853. 


CONTENTS. 


CHAPTER  I. 

Page. 

Nature  and  Design  of  this  Work, 1 

CHAPTER  II. 

Signs  and  their  Laws, 24 

CHAPTER  IH. 

Derivation  of  the  Laws, 39 

CHAPTER  IV. 

Division  of  Propositions, 52 

CHAPTER  V. 

Principles  of  Symbolical  Reasoning, 66 

CHAPTER  VI. 

Of  Interpretation, 80 

CHAPTER  VTI. 

Of  Elimination, 99 

' CHAPTER  VIH. 

Of  Reduction, 114 

CHAPTER  IX. 

Methods  of  Abbreviation, 130 


CONTENTS. 


Page. 

CHAPTER  X. 

Conditions  of  a Perfect  Method, ...  150 

CHAPTER  XI. 

Of  Secondary  Propositions, 159 

CHAPTER  XII. 

Methods  in  Secondary  Propositions, 177 

CHAPTER  XIII. 

Clarke  and  Spinoza, 185 

CHAPTER  XIV. 

Example  of  Analysis, 219 

CHAPTER  XV. 

Of  the  Aristotelian  Logic, 226 

CHAPTER  XVI. 

Of  the  Theory  of  Probabilities, 243 

CHAPTER  XVII. 

General  Method  in  Probabilities, 253 

CHAPTER  XVIII. 

Elementary  Illustrations, 276 

CHAPTER  XIX. 

Of  Statistical  Conditions, 295 

CHAPTER  XX. 

Problems  on  Causes, 320 

CHAPTER  XXI. 

Probability  of  Judgments, 376 

CHAPTER  XXII. 

Constitution  of  the  Intellect, 399 


NOTE. 


In  Prop.  II.,  p.  261,  by  the  “ absolute  probabilities”  of  the  events  x,  y,  z . . is 
meant  simply  what  the  probabilities  of  those  events  ought  to  be,  in  order  that, 
regarding  them  as  independent,  and  their  probabilities  as  our  only  data,  the  cal- 
culated probabilities  of  the  same  events  under  the  condition  V should  be  p,  q,  r . . 
The  statement  of  the  appended  problem  of  the  urn  must  be  modified  in  a similar 
way.  The  true  solution  of  that  problem,  as  actually  stated,  is  p = cp,  q = cq, 
in  which  c is  the  arbitrary  probability  of  the  condition  that  the  ball  drawn  shall 
be  either  white,  or  of  marble,  or  both  at  once See  p.  270,  Case  II. 

Accordingly,  since  by  the  logical  reduction  the  solution  of  all  questions  in 
the  theory  of  probabilities  is  brought  to  a form  in  which,  from  the  probabilities 
of  simple  events,  s,  f,  &c.  under  a given  condition,  V,  it  is  required  to  determine 
the  probability  of  some  combination,  A,  of  those  events  under  the  same  condi- 
tion, the  principle  of  the  demonstration  in  Prop.  IV.  is  really  the  following  : — 
“ The  probability  of  such  combination  A under  the  condition  V must  be  calcu- 
lated as  if  the  events  s,  f,  &c.  were  independent,  and  possessed  of  such  probabi- 
lities as  would  cause  the  derived  probabilities  of  the  said  events  under  the  same 
condition  V to  be  such  as  are  assigned  to  them  in  the  data.”  This  principle  I 
regard  as  axiomatic.  At  the  same  time  it  admits  of  indefinite  verification,  as 
well  directly  as  through  the  results  of  the  method  of  which  it  forms  the  basis. 
I think  it  right  to  add,  that  it  was  in  the  above  form  that  the  principle  first  pre- 
sented itself  to  my  mind,  and  that  it  is  thus  that  I have  always  understood  it, 
the  error  in  the  particular  problem  referred  to  having  arisen  from  inadvertence 
in  the  choice  of  a material  illustration. 


AN  INVESTIGATION 

OF 


THE  LAWS  OF  THOUGHT. 


AN  INVESTIGATION 


OF 

THE  LAWS  OE  THOUGHT. 


CHAPTER  I. 

NATURE  AND  DESIGN  OF  THIS  WORK. 

1 . r I ''HE  design  of  the  following  treatise  is  to  investigate  the 
fundamental  laws  of  those  operations  of  the  mind  by  which 
reasoning  is  performed;  to  give  expression  to  them  in  the  symboli- 
cal language  of  a Calculus,  and  upon  this  foundation  to  establish  the 
science  of  Logic  and  construct  its  method  ; to  make  that  method 
itself  the  basis  of  a general  method  for  the  application  of  the  ma- 
thematical doctrine  of  Probabilities ; and,  finally,  to  collect  from 
the  various  elements  of  truth  brought  to  view  in  the  course  of 
these  inquiries  some  probable  intimations  concerning  the  nature 
and  constitution  of  the  human  mind. 

2.  That  this  design  is  not  altogether  a novel  one  it  is  almost 
needless  to  remark,  and  it  is  well  known  that  to  its  two  main 
practical  divisions  of  Logic  and  Probabilities  a very  considerable 
share  of  the  attention  of  philosophers  has  been  directed.  In  its 
ancient  and  scholastic  form,  indeed,  the  subject  of  Logic  stands 
almost  exclusively  associated  with  the  great  name  of  Aristotle. 
As  it  was  presented  to  ancient  Greece  in  the  partly  technical, 
partly  metaphysical  disquisitions  of  the  Organon,  such,  with 
scarcely  any  essential  change,  it  has  continued  to  the  present 
day.  The  stream  of  original  inquiry  has  rather  been  directed 
towards  questions  of  general  philosophy,  which,  though  they 


2 NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

have  arisen  among  the  disputes  of  the  logicians,  have  outgrown 
their  origin,  and  given  to  successive  ages  of  speculation  their  pe- 
culiar bent  and  character.  The  eras  of  Porphyry  and  Proclus, 
of  Anselm  and  Abelard,  of  Ramus,  and  of  Descartes,  together 
with  the  final  protests  of  Bacon  and  Locke,  rise  up  before  the 
mind  as  examples  of  the  remoter  influences  of  the  study  upon  the 
course  of  human  thought,  partly  in  suggesting  topics  fertile  of 
discussion,  partly  in  provoking  remonstrance  against  its  own  un- 
due pretensions.  The  history  of  the  theory  of  Probabilities,  on 
the  other  hand,  has  presented  far  more  of  that  character  of  steady- 
growth  which  belongs  to  science.  In  its  origin  the  early  genius 
of  Pascal, — in  its  maturer  stages  of  development  the  most  recon- 
dite of  all  the  mathemat  ical  speculations  of  Laplace, — were  direct- 
ed to  its  improvement ; to  omit  here  the  mention  of  other  names 
scarcely  less  distinguished  than  these.  As  the  study  of  Logic  has 
been  remarkable  for  the  kindred  questions  of  Metaphysics  to 
which  it  has  given  occasion,  so  that  of  Probabilities  also  has  been 
remarkable  for  the  impulse  which  it  has  bestowed  upon  the 
higher  departments  of  mathematical  science.  Each  of  these  sub- 
jects has,  moreover,  been  justly  regarded  as  having  relation  to  a 
speculative  as  well  as  to  a practical  end.  To  enable  us  to  deduce 
correct  inferences  from  given  premises  is  not  the  only  object  of 
Logic  ; nor  is  it  the  sole  claim  of  the  theory  of  Probabilities  that 
it  teaches  us  how  to  establish  the  business  of  life  assurance  on  a 
secure  basis  ; and  how  to  condense  whatever  is  valuable  in  the 
records  of  innumerable  observations  in  astronomy,  in  physics,  or 
in  that  field  of  social  inquiry  which  is  fast  assuming  a character 
of  great  importance.  Both  these  studies  have  also  an  interest 
of  another  kind,  derived  from  the  light  which  they  shed  upon 
the  intellectual  powers.  They  instruct  us  concerning  the  mode 
in  which  language  and  number  serve  as  instrumental  aids  to  the 
processes  of  reasoning ; they  reveal  to  us  in  some  degree  the 
connexion  between  different  powers  of  our  common  intellect ; 
they  set  before  us  what,  in  the  two  domains  of  demonstrative  and 
of  probable  knowledge,  are  the  essential  standards  of  truth  and 
correctness, — standards  not  derived  from  without,  but  deeply 
founded  in  the  constitution  of  the  human  faculties.  These  ends 
of  speculation  yield  neither  in  interest  nor  in  dignity,  nor  yet,  it 


3 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

may  be  added,  in  importance,  to  the  practical  objects,  with  the 
pursuit  of  which  they  have  been  historically  associated.  To  un- 
fold the  secret  laws  and  relations  of  those  high  faculties  of 
thought  by  which  all  beyond  the  merely  perceptive  knowledge 
of  the  world  and  of  ourselves  is  attained  or  matured,  is  an  object 
which  does  not  stand  in  need  of  commendation  to  a rational 
mind. 

3.  But  although  certain  parts  of  the  design  of  this  work  have 
been  entertained  by  others,  its  general  conception,  its  method, 
and,  to  a considerable  extent,  its  results,  are  believed  to  be  ori- 
ginal. For  this  reason  I shall  offer,  in  the  present  chapter,  some 
preparatory  statements  and  explanations,  in  order  that  the  real 
aim  of  this  treatise  may  be  understood,  and  the  treatment  of  its 
subject  facilitated. 

It  is  designed,  in  the  first  place,  to  investigate  the  fundamen- 
tal laws  of  those  operations  of  the  mind  by  which  reasoning  is 
performed.  It  is  unnecessary  to  enter  here  into  any  argument  to 
prove  that  the  operations  of  the  mind  are  in  a certain  real  sense 
subject  to  laws,  and  that  a science  of  the  mind  is  therefore  possible. 
If  these  are  questions  which  admit  of  doubt,  that  doubt  is  not 
to  be  met  by  an  endeavour  to  settle  the  point  of  dispute  a priori, 
but  by  directing  the  attention  of  the  objector  to  the  evidence  of 
actual  laws,  by  referring  him  to  an  actual  science.  And  thus  the 
solution  of  that  doubt  would  belong  not  to  the  introduction  to 
this  treatise,  but  to  the  treatise  itself.  Let  the  assumption  be 
granted,  that  a science  of  the  intellectual  powers  is  possible,  and 
let  us  for  a moment  consider  how  the  knowledge  of  it  is  to  be 
obtained. 

4.  Like  all  other  sciences,  that  of  the  intellectual  operations 
must  primarily  rest  upon  observation, — the  subject  of  such  ob- 
servation being  the  very  operations  and  processes  of  which  we 
desire  to  determine  the  laws.  But  while  the  necessity  of  a foun- 
dation in  experience  is  thus  a condition  common  to  all  sciences, 
there  are  some  special  differences  between  the  modes  in  which 
this  principle  becomes  available  for  the  determination  of  general 
truths  when  the  subject  of  inquiry  is  the  mind,  and  when  the 
subject  is  external  nature.  To  these  it  is  necessary  to  direct 
attention. 


4 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

The  general  laws  of  Nature  are  not,  for  the  most  part,  imme- 
diate objects  of  perception.  They  are  either  inductive  inferences 
from  a large  body  of  facts,  the  common  truth  in  which  they  ex- 
press, or,  in  their  origin  at  least,  physical  hypotheses  of  a causal 
nature  serving  to  explain  phasnomena  with  undeviating  precision, 
and  to  enable  us  to  predict  new  combinations  of  them.  They 
are  in  all  cases,  and  in  the  strictest  sense  of  the  term,  probable 
conclusions,  approaching,  indeed,  ever  and  ever  nearer  to  cer- 
tainty, as  they  receive  more  and  more  of  the  confirmation  of  ex- 
perience. But  of  the  character  of  probability,  in  the  strict  and 
proper  sense  of  that  term,  they  are  never  wholly  divested.  On  the 
other  hand,  the  knowledge  of  the  laws  of  the  mind  does  not  require 
as  its  basis  any  extensive  collection  of  observations.  The  general 
truth  is  seen  in  the  particular  instance,  and  it  is  not  confirmed 
by  the  repetition  of  instances.  We  may  illustrate  this  position 
by  an  obvious  example.  It  may  be  a question  whether  that  for- 
mula of  reasoning,  which  is  called  the  dictum  of  Aristotle,  de  omni 
et  nullo,  expresses  a primary  law  of  human  reasoning  or  not ; but 
it  is  no  question  that  it  expresses  a general  truth  in  Logic.  Now 
that  truth  is  made  manifest  in  all  its  generality  by  reflection 
upon  a single  instance  of  its  application.  And  this  is  both  an 
evidence  that  the  particular  principle  or  formula  in  question  is 
founded  upon  some  general  law  or  laws  of  the  mind,  and  an  illus- 
tration of  the  doctrine  that  the  perception  of  such  general  truths 
is  not  derived  from  an  induction  from  many  instances,  but  is  in- 
volved in  the  clear  apprehension  of  a single  instance.  In  con- 
nexion with  this  truth  is  seen  the  not  less  important  one  that 
our  knowledge  of  the  laws  upon  which  the  science  of  the  intellec- 
tual powers  rests,  whatever  may  be  its  extent  or  its  deficiency,  is 
not  probable  knowledge.  For  we  not  only  see  in  the  particular 
example  the  general  truth,  but  we  see  it  also  as  a certain  truth, — 
a truth,  our  confidence  in  which  will  not  continue  to  increase 
with  increasing  experience  of  its  practical  verifications. 

5.  But  if  the  general  truths  of  Logic  are  of  such  a nature  that 
when  presented  to  the  mind  they  at  once  command  assent, 
wherein  consists  the  difficulty  of  constructing  the  Science  of 
Logic  ? Not,  it  may  be  answered,  in  collecting  the  materials  of 
knowledge,  but  in  discriminating  their  nature,  and  determining 


5 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

their  mutual  place  and  relation.  All  sciences  consist  of  general 
truths,  but  of  those  truths  some  only  are  primary  and  fundamen- 
tal, others  are  secondary  and  derived.  The  laws  of  elliptic  mo- 
tion, discovered  by  Kepler,  are  general  truths  in  astronomy,  but 
they  are  not  its  fundamental  truths.  And  it  is  so  also  in  the 
purely  mathematical  sciences.  An  almost  boundless  diversity  of 
theorems,  which  are  known,  and  an  infinite  possibility  of  others, 
as  yet  unknown,  rest  together  upon  the  foundation  of  a few  sim- 
ple axioms ; and  yet  these  are  all  general  truths.  It  may  be 
added,  that  they  are  truths  which  to  an  intelligence  sufficiently 
refined  would  shine  forth  in  their  own  unborrowed  light,  with- 
out the  need  of  those  connecting  links  of  thought,  those  steps 
of  wearisome  and  often  painful  deduction,  by  which  the  know- 
ledge of  them  is  actually  acquired.  Let  us  define  as  fundamental 
those  laws  and  principles  from  which  all  other  general  truths  of 
science  may  be  deduced,  and  into  which  they  may  all  be  again 
resolved.  Shall  we  then  err  in  regarding  that  as  the  true  science 
of  Logic  which,  laying  down  certain  elementary  laws,  confirmed 
by  the  very  testimony  of  the  mind,  permits  us  thence  to  deduce, 
by  uniform  processes,  the  entire  chain  of  its  secondary  conse- 
quences, and  furnishes,  for  its  practical  applications,  methods  of 
perfect  generality  ? Let  it  be  considered  whether  in  any  science, 
viewed  either  as  a system  of  truth  or  as  the  foundation  of  a prac- 
tical art,  there  can  properly  be  any  other  test  of  the  completeness 
and  the  fundamental  character  of  its  laws,  than  the  completeness 
of  its  system  of  derived  truths,  and  the  generality  of  the  methods 
which  it  serves  to  establish.  Other  questions  may  indeed  pre- 
sent themselves.  Convenience,  prescription,  individual  prefe- 
rence, may  urge  their  claims  and  deserve  attention.  But  as 
respects  the  question  of  what  constitutes  science  in  its  abstract 
integrity,  I apprehend  that  qo  other  considerations  than  the 
above  are  properly  of  any  value. 

6.  It  is  designed,  in  the  next  place,  to  give  expression  in  this 
treatise  to  the  fundamental  laws  of  reasoning  in  the  symbolical 
language  of  a Calculus.  Upon  this  head  it  will  suffice  to  say,  that 
those  laws  are  such  as  to  suggest  this  mode  of  expression,  and 
to  give  to  it  a peculiar  and  exclusive  fitness  for  the  ends  in  view. 


6 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 


There  is  not  only  a close  analogy  between  the  operations  of  the 
mind  in  general  reasoning  and  its  operations  in  the  particular 
science  of  Algebra,  but  there  is  to  a considerable  extent  an  exact 
agreement  in  the  laws  by  which  the  two  classes  of  operations  are 
conducted.  Of  course  the  laws  must  in  both  cases  be  determined 
independently ; any  formal  agreement  between  them  can  only  be 
established  a posteriori  by  actual  comparison.  To  borrow  the 
notation  of  the  science  of  Number,  and  then  assume  that  in  its 
new  application  the  laws  by  which  its  use  is  governed  will  remain 
unchanged,  would  be  mere  hypothesis.  There  exist,  indeed, 
certain  general  principles  founded  in  the  very  nature  of  language, 
by  which  the  use  of  symbols,  which  are  but  the  elements  of 
scientific  language,  is  determined.  To  a certain  extent  these 
elements  are  arbitrary.  Their  interpretation  is  purely  conven- 
tional : we  are  permitted  to  employ  them  in  whatever  sense  we 
please.  But  this  permission  is  limited  by  two  indispensable  con- 
ditions,— first,  that  from  the  sense  once  conventionally  established 
we  never,  in  the  same  process  of  reasoning,  depart ; secondly, 
that  the  laws  by  which  the  process  is  conducted  be  founded  ex- 
clusively upon  the  above  fixed  sense  or  meaning  of  the  symbols 
employed.  In  accordance  with  these  principles,  any  agreement 
which  may  be  established  between  the  laws  of  the  symbols  of 
Logic  and  those  of  Algebra  can  but  issue  in  an  agreement  of  pro- 
cesses. The  two  provinces  of  interpretation  remain  apart  and 
independent,  each  subject  to  its  own  laws  and  conditions. 

Now  the  actual  investigations  of  the  following  pages  exhibit 
Logic,  in  its  practical  aspect,  as  a system  of  processes  carried  on 
by  the  aid  of  symbols  having  a definite  interpretation,  and  sub- 
ject to  laws  founded  upon  that  interpretation  alone.  But  at  the 
same  time  they  exhibit  those  laws  as  identical  in  form  with  the 
laws  of  the  general  symbols  of  algebra,  with  this  single  addition, 
viz.,  that  the  symbols  of  Logic  are  further  subject  to  a special 
law  (Chap,  ii.),  to  which  the  symbols  of  quantity,  as  such,  are 
not  subject.  Upon  the  nature  and  the  evidence  of  this  law  it  is  not 
purposed  here  to  dwell.  These  questions  will  be  fully  discussed 
in  a future  page.  But  as  constituting  the  essential  ground  of 
difference  between  those  forms  of  inference  with  which  Logic  is 


7 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

conversant,  and  those  which  present  themselves  in  the  particular 
science  of  Number,  the  law  in  question  is  deserving  of  more 
than  a passing  notice.  It  may  be  said  that  it  lies  at  the  very 
foundation  of  general  reasoning, — that  it  governs  those  intellec- 
tual acts  of  conception  or  of  imagination  which  are  preliminary  to 
the  processes  of  logical  deduction,  and  that  it  gives  to  the  pro- 
cesses themselves  much  of  their  actual  form  and  expression.  It 
may  hence  be  affirmed  that  this  law  constitutes  the  germ  or  semi- 
nal principle,  of  which  every  approximation  to  a general  method 
in  Logic  is  the  more  or  less  perfect  development. 

7.  The  principle  has  already  been  laid  down  (5)  that  the 
sufficiency  and  truly  fundamental  character  of  any  assumed  sys- 
tem of  laws  in  the  science  of  Logic  must  partly  be  seen  in  the 
perfection  of  the  methods  to  which  they  conduct  us.  It  remains, 
then,  to  consider  what  the  requirements  of  a general  method  in 
Logic  are,  and  how  far  they  are  fulfilled  in  the  system  of  the  pre- 
sent work. 

Logic  is  conversant  with  two  kinds  of  relations, — relations 
among  things,  and  relations  among  facts.  But  as  facts  are  ex- 
pressed by  propositions,  the  latter  species  of  relation  may,  at 
least  for  the  purposes  of  Logic,  be  resolved  into  a relation  among 
propositions.  The  assertion  that  the  fact  or  event  A is  an  inva- 
riable consequent  of  the  fact  or  event  B may,  to  this  extent  at 
least,  be  regarded  as  equivalent  to  the  assertion,  that  the  truth 
of  the  proposition  affirming  the  occurrence  of  the  event  B always 
implies  the  truth  of  the  proposition  affirming  the  occurrence  of 
the  event  A.  Instead,  then,  of  saying  that  Logic  is  conversant 
with  relations  among  things  and  relations  among  facts,  we  are 
permitted  to  say  that  it  is  concerned  with  relatione  among  things 
and  relations  among  propositions.  Of  the  former  kind  of  relations 
we  have  an  example  in  the  proposition — “ All  men  are  mortal 
of  the  latter  kind  in  the  proposition — “ If  the  sun  is  totally 
eclipsed,  the  stars  will  become  visible.”  The  one  expresses  a re- 
lation between  “men”  and  “ mortal  beings,”  the  other  between 
the  elementary  propositions — “The  sun  is  totally  eclipsed;” 
“ The  stars  will  become  visible.”  Among  such  relations  I sup- 
pose to  be  included  those  which  affirm  or  deny  existence  with 
respect  to  things,  and  those  which  affirm  or  deny  truth  with  re- 


8 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

spect  to  propositions.  Now  let  those  things  or  those  propositions 
among  which  relation  is  expressed  be  termed  the  elements  of 
the  propositions  by  which  such  relation  is  expressed.  Proceed- 
ing from  this  definition,  we  may  then  say  that  the  premises  of  any 
logical  argument  express  given  relations  among  certain  elements, 
and  that  the  conclusion  must  express  an  implied  relation  among 
those  elements,  or  among  a part  of  them,  i.  e.  a relation  implied 
by  or  inferentially  involved  in  the  premises. 

8.  Now  this  being  premised,  the  requirements  of  a general 
method  in  Logic  seem  to  be  the  following : — 

1st.  As  the  conclusion  must  express  a relation  among  the 
whole  or  among  a part  of  the  elements  involved  in  the  premises, 
it  is  requisite  that  we  .should  possess  the  means  of  eliminating 
those  elements  which  we  desire  not  to  appear  in  the  conclusion, 
and  of  determining  the  whole  amount  of  relation  implied  by  the 
premises  among  the  elements  which  we  wish  to  retain.  Those 
elements  which  do  not  present  themselves  in  the  conclusion  are, 
in  the  language  of  the  common  Logic,  called  middle  terms ; and 
the  species  of  elimination  exemplified  in  treatises  on  Logic  consists 
in  deducing  from  two  propositions,  containing  a common  element 
or  middle  term,  a conclusion  connecting  the  two  remaining  terms. 
But  the  problem  of  elimination,  as  contemplated  in  this  work, 
possesses  a much  wider  scope.  It  proposes  not  merely  the  elimi- 
nation of  one  middle  term  from  two  propositions,  but  the  elimi- 
nation generally  of  middle  terms  from  propositions,  without 
regard  to  the  number  of  either  of  them,  or  to  the  nature  of  .their 
connexion.  To  this  object  neither  the  processes  of  Logic  nor 
those  of  Algebra,  in  their  actual  state,  present  any  strict  parallel. 
In  the  latter  science  the  problem  of  elimination  is  known  to  be 
limited  in  the  following  manner : — From  two  equations  we  can 
eliminate  one  symbol  of  quantity ; from  three  equations  two 
symbols  ; and,  generally,  from  n equations  n - 1 symbols.  But 
though  this  condition,  necessary  in  Algebra,  seems  to  prevail  in 
the  existing  Logic  also,  it  has  no  essential  place  in  Logic  as  a 
science.  There,  no  relation  whatever  can  be  proved  to  prevail 
between  the  number  of  terms  to  be  eliminated  and  the  number 
of  propositions  from  which  the  elimination  is  to  be  effected. 
From  the  equation  representing  a single  proposition,  any  num- 


9 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

ber  of  symbols  representing  terms  or  elements  in  Logic  may  be 
eliminated ; and  from  any  number  of  equations  representing  pro- 
positions, one  or  any  other  number  of  symbols  of  this  kind  may 
be  eliminated  in  a similar  manner.  For  such  elimination  there 
exists  one  general  process  applicable  to  all  cases.  This  is  One  of 
the  many  remarkable  consequences  of  that  distinguishing  law  of 
the  symbols  of  Logic,  to  which  attention  has  been  already 
directed. 

2ndly.  It  should  be  within  the  province  of  a general  method 
in  Logic  to  express  the  final  relation  among  the  elements  of  the 
conclusion  by  any  admissible  kind  of  proposition,  or  in  any  se- 
lected order  of  terms.  Among  varieties  of  kind  we  may  reckon 
those  which  logicians  have  designated  by  the  terms  categorical, 
hypothetical,  disjunctive,  &c.  To  a choice  or  selection  in  the 
order  of  the  terms,  we  may  refer  whatsoever  is  dependent  upon 
the  appearance  of  particular  elements  in  the  subject  or  in  the 
predicate,  in  the  antecedent  or  in  the  consequent,  of  that  propo- 
sition which  forms  the  “ conclusion.”  But  waiving  the  language 
of  the  schools,  let  us  consider  what  really  distinct  species  of 
problems  may  present  themselves  to  our  notice.  We  have  seen 
that  the  elements  of  the  final  or  inferred  relation  may  either  be 
things  or  propositions.  Suppose  the  former  case ; then  it  might 
be  required  to  deduce  from  the  premises  a definition  or  description 
of  some  one  thing,  or  class  of  things,  constituting  an  element  of 
the  conclusion  in  terms  of  the  other  things  involved  in  it.  Or 
we  might  form  the  conception  of  some  thing  or  class  of  things, 
involving  more  than  one  of  the  elements  of  the  conclusion,  and 
require  its  expression  in  terms  of  the  other  elements.  Again, 
suppose  the  elements  retained  in  the  conclusion  to  be  propo- 
sitions, we  might  desire  to  ascertain  such  points  as  the  following, 
viz.,  Whether,  in  virtue  of  the  premises,  any  of  those  propo- 
sitions, taken  singly,  are  true  or  false  ? — Whether  particular 
combinations  of  them  are  true  or  false  ? — Whether,  assuming  a 
particular  proposition  to  be  true,  any  consequences  will  follow, 
and  if  so,  what  consequences,  with  respect  to  the  other  propo- 
sitions ? — Whether  any  particular  condition  being  assumed  with 
reference  to  certain  of  the  propositions,  any  consequences,  and 
what  consequences,  will  follow  with  respect  to  the  others  ? and 


10 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

so  on.  I say  that  these  are  general  questions,  which  it  should 
fall  within  the  scope  or  province  of  a general  method  in  Logic  to 
solve.  Perhaps  we  might  include  them  all  under  this  one  state- 
ment of  the  final  problem  of  practical  Logic.  Given  a set  of 
premises  expressing  relations  among  certain  elements,  whether 
things  or  propositions : required  explicitly  the  whole  relation 
consequent  among  any  of  those  elements  under  any  proposed 
conditions,  and  in  any  proposed  form.  That  this  problem,  under 
all  its  aspects,  is  resolvable,  will  hereafter  appear.  But  it  is  not 
for  the  sake  of  noticing  this  fact,  that  the  above  inquiry  into  the 
nature  and  the  functions  of  a general  method  in  Logic  has  been 
introduced.  It  is  necessary  that  the  reader  should  apprehend 
what  are  the  specific  ends  of  the  investigation  upon  which  we 
are  entering,  as  well  as  the  principles  which  are  to  guide  us  to 
the  attainment  of  them. 

9.  Possibly  it  may  here  be  said  that  the  Logic  of  Aristotle, 
in  its  rules  of  syllogism  and  conversion,  sets  forth  the  elementary 
processes  of  which  all  reasoning  consists,  and  that  beyond  these 
there  is  neither  scope  nor  occasion  for  a general  method.  I have 
no  desire  to  point  out  the  defects  of  the  common  Logic,  nor  do  I 
wish  to  refer  to  it  any  further  than  is  necessary,  in  order  to  place 
in  its  true  light  the  nature  of  the  present  treatise.  With  this 
end  alone  in  view,  I would  remark  : — 1st.  That  syllogism,  con- 
version, &c.,  are  not  the  ultimate  processes  of  Logic.  It  will 
be  shown  in  this  treatise  that  they  are  founded  upon,  and  are  re- 
solvable into,  ulterior  and  more  simple  processes  which  constitute 
the  real  elements  of  method  in  Logic.  Nor  is  it  true  in  fact  that 
all  inference  is  reducible  to  the  particular  forms  of  syllogism  and 
conversion. — Vide  Chap.  xv.  2ndly.  If  all  inference  were  re- 
ducible to  these  two  processes  (and  it  has  been  maintained  that 
it  is  reducible  to  syllogism  alone),  there  would  still  exist  the 
same  necessity  for  a general  method.  For  it  would  still  be  re- 
quisite to  determine  in  what  order  the  processes  should  succeed 
each  other,  as  well  as  their  particular  nature,  in  order  that  the 
desired  relation  should  be  obtained.  By  the  desired  relation  I 
mean  that  full  relation  which,  in  virtue  of  the  premises,  connects 
any  elements  selected  out  of  the  premises  at  will,  and  which, 
moreover,  expresses  that  relation  in  any  desired  form  and  order. 


CHAP.  1.1  NATURE  AND  DESIGN  OF  THIS  WORK.  1 1 

If  we  may  judge  from  the  mathematical  sciences,  which  are  the 
most  perfect  examples  of  method  known,  this  directive  function 
of  Method  constitutes  its  chief  office  and  distinction.  The  fun- 
damental processes  of  arithmetic,  for  instance,  are  in  themselves 
but  the  elements  of  a possible  science.  To  assign  their  nature  is 
the  first  business  of  its  method,  but  to  arrange  their  succession 
is  its  subsequent  and  higher  function.  In  the  more  complex 
examples  of  logical  deduction,  and  especially  in  those  which  form 
a basis  for  the  solution  of  difficult  questions  in  the  theory  of 
Probabilities,  the  aid  of  a directive  method,  such  as  a Calculus 
alone  can  supply,  is  indispensable. 

10.  Whence  it  is  that  the  ultimate  laws  of  Logic  are  mathe- 
matical in  their  form ; why  they  are,  except  in  a single  point, 
identical  with  the  general  laws  of  Number ; and  why  in  that  par- 
ticular point  they  differ ; — are  questions  upon  which  it  might  not 
be  very  remote  from  presumption  to  endeavour  to  pronounce  a 
positive  judgment.  Probably  they  lie  beyond  the  reach  of  our 
limited  faculties.  It  may,  perhaps,  be  permitted  to  the  mind  to 
attain  a knowledge  of  the  laws  to  which  it  is  itself  subject,  with- 
out its  being  also  given  to  it  to  understand  their  ground  and 
origin,  or  even,  except  in  a very  limited  degree,  to  comprehend 
their  fitness  for  their  end,  as  compared  with  other  and  conceivable 
systems  of  law.  Such  knowledge  is,  indeed,  unnecessary  for  the 
ends  of  science,  which  properly  concerns  itself  with  what  is,  and 
seeks  not  for  grounds  of  preference  or  reasons  of  appointment. 
These  considerations  furnish  a sufficient  answer  to  all  protests 
against  the  exhibition  of  Logic  in  the  form  of  a Calculus.  It  is 
not  because  we  choose  to  assign  to  it  such  a mode  of  manifes- 
tation, but  because  the  ultimate  laws  of  thought  render  that  mode 
possible,  and  prescribe  its  character,  and  forbid,  as  it  would 
seem,  the  perfect  manifestation  of  the  science  in  any  other  form, 
that  such  a mode  demands  adoption.  It  is  to  be  remembered 
that  it  is  the  business  of  science  not  to  create  laws,  but  to  discover 
them.  We  do  not  originate  the  constitution  of  our  own  minds, 
greatly  as  it  may  be  in  our  power  to  modify  their  character. 
And  as  the  laws  of  the  human  intellect  do  not  depend  upon  our 
will,  so  the  forms  of  the  science,  of  which  they  constitute  the  ba- 
sis, are  in  all  essential  regards  independent  of  individual  choice. 


12  NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

1 1 . Beside  the  general  statement  of  the  principles  of  the 
above  method,  this  treatise  will  exhibit  its  application  to  the 
analysis  of  a considerable  variety  of  pi’opositions,  and  of  trains  of 
propositions  constituting  the  premises  of  demonstrative  argu- 
ments. These  examples  have  been  selected  from  various  writers, 
they  differ  greatly  in  complexity,  and  they  embrace  a wide  range 
of  subjects.  Though  in  this  particular  respect  it  may  appear  to 
some  that  too  great  a latitude  of  choice  has  been  exercised,  I do 
not  deem  it  necessary  to  offer  any  apology  upon  this  account. 
That  Logic,  as  a science,  is  susceptible  of  very  wide  applications 
is  admitted ; but  it  is  equally  certain  that  its  ultimate  forms  and 
processes  are  mathematical.  Any  objection  a priori  which  may 
therefore  be  supposed  to  lie  against  the  adoption  of  such  forms 
and  processes  in  the  discussion  of  a problem  of  morals  or  of  ge- 
neral philosophy  must  be  founded  upon  misapprehension  or  false 
analogy.  It  is  not  of  the  essence  of  mathematics  to  be  conversant 
with  the  ideas  of  number  and  quantity.  Whether  as  a general 
habit  of  mind  it  would  be  desirable  to  apply  symbolical  processes 
to  moral  argument,  is  another  question.  Possibly,  as  I have 
elsewhere  observed,*  the  perfection  of  the  method  of  Logic  may 
be  chiefly  valuable  as  an  evidence  of  the  speculative  truth  of  its 
principles.  To  supersede  the  employment  of  common  reasoning, 
or  to  subject  it  to  the  rigour  of  technical  forms,  would  be  the  last 
desire  of  one  who  knows  the  value  of  that  intellectual  toil  and 
warfare  which  imparts  to  the  mind  an  athletic  vigour,  and  teaches 
it  to  contend  with  difficulties,  and  to  rely  upon  itself  in  emer- 
gencies. Nevertheless,  cases  may  arise  in  which  the  value  of  a 
scientific  procedure,  even  in  those  things  which  fall  confessedly 
under  the  ordinary  dominion  of  the  reason,  may  be  felt  and  ac- 
knowledged. Some  examples  of  this  kind  will  be  found  in  the 
present  work. 

12.  The  general  doctrine  and  method  of  Logic  above  ex- 
plained form  also  the  basis  of  a theory  and  corresponding  method 
of  Probabilities.  Accordingly,  the  development  of  such  a theory 
and  method,  upon  the  above  principles,  will  constitute  a distinct 
object  of  the  present  treatise.  Of  the  nature  of  this  application 
it  may  be  desirable  to  give  here  some  account,  more  especially  as 

* Mathematical  Analysis  of  Logic.  London : G.  Bell.  1847- 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK.  13 

regards  the  character  of  the  solutions  to  which  it  leads.  In  con- 
nexion with  this  object  some  further  detail  will  be  requisite  con- 
cerning the  forms  in  which  the  results  of  the  logical  analysis  are 
presented. 

The  ground  of  this  necessity  of  a prior  method  in  Logic,  as 
the  basis  of  a theory  of  Probabilities,  may  be  stated  in  a few 
words.  Before  we  can  determine  the  mode  in  which  the  expected 
frequency  of  occurrence  of  a particular  event  is  dependent  upon 
the  known  frequency  of  occurrence  of  any  other  events,  we  must  be 
acquainted  with  the  mutual  dependence  of  the  events  themselves. 
Speaking  technically,  we  must  be  able  to  express  the  event 
whose  probability  is  sought,  as  a function  of  the  events  whose 
probabilities  are  given.  Now  this  explicit  determination  belongs 
in  all  instances  to  the  department  of  Logic.  Probability,  how- 
ever, in  its  mathematical  acceptation,  admits  of  numerical  mea- 
surement. Hence  the  subject  of  Probabilities  belongs  equally  to 
the  science  of  Number  and  to  that  of  Logic.  In  recognising  the 
co-ordinate  existence  of  both  these  elements,  the  present  treatise 
differs  from  all  previous  ones  ; and  as  this  difference  not  only 
affects  the  question  of  the  possibility  of  the  solution  of  problems 
in  a large  number  of  instances,  but  also  introduces  new  and  im- 
portant elements  into  the  solutions  obtained,  I deem  it  necessary 
to  state  here,  at  some  length,  the  peculiar  consequences  of  the 
theory  developed  in  the  following  pages. 

13.  The  measure  of  the  probability  of  an  event  is  usually 
defined  as  a fraction,  of  which  the  numerator  represents  the  num- 
ber of  cases  favourable  to  the  event,  and  the  denominator  the 
whole  number  of  cases  favourable  and  unfavourable ; all  cases 
being  supposed  equally  likely  to  happen.  That  definition  is 
adopted  in  the  present  work.  At  the  same  time  it  is  shown  that 
there  is  another  aspect  of  the  subject  (shortly  to  be  referred  to) 
which  might  equally  be  regarded  as  fundamental,  and  which 
would  actually  lead  to  the  same  system  of  methods  and  conclu- 
sions. It  may  be  added,  that  so  far  as  the  received  conclusions 
of  the  theory  of  Probabilities  extend,  and  so  far  as  they  are  con- 
sequences of  its  fundamental  definitions,  they  do  not  differ  from 
the  results  (supposed  to  be  equally  correct  in  inference)  of  the 
method  of  this  work. 


14 


NATURE  AND  DESIGN  OF  THIS  WORK.  TcHAP.  I. 

Again,  although  questions  in  the  theory  of  Probabilities 
present  themselves  under  various  aspects,  and  may  be  variously 
modified  by  algebraical  and  other  conditions*  there  seems  to  be 
one  general  type  to  which  all  such  questions,  or  so  much  of  each 
of  them  as  truly  belongs  to  the  theory  of  Probabilities,  may  be 
referred.  Considered  with  reference  to  the  data  and  the  qucesi- 
turn , that  type  may  be  described  as  follows : — 1st.  The  data  are 
the  probabilities  of  one  or  more  given  events,  each  probability 
being  either  that  of  the  absolute  fulfilment  of  the  event  to  which 
it  relates,  or  the  probability  of  its  fulfilment  under  given  sup- 
posed conditions.  2ndly.  The  qucesitum,  or  object  sought,  is  the 
probability  of  the  fulfilment,  absolutely  or  conditionally,  of  some 
other  event  differing  in  expression  from  those  in  the  data,  but 
more  or  less  involving  the  same  elements.  As  concerns  the  data, 
they  are  either  causally  given , — as  when  the  probability  of  a par- 
ticular throw  of  a die  is  deduced  from  a knowledge  of  the  consti- 
tution of  the  piece, — or  they  are  derived  from  observation  of 
repeated  instances  of  the  success  or  failure  of  events.  In  the 
latter  case  the  probability  of  an  event  may  be  defined  as  the 
limit  toward  which  the  ratio  of  the  favourable  to  the  whole  num- 
ber of  observed  cases  approaches  (the  uniformity  of  nature  being 
presupposed)  as  the  observations  are  indefinitely  continued. 
Lastly,  as  concerns  the  nature  or  relation  of  the  events  in  ques- 
tion, an  important  distinction  remains.  Those  events  are  either 
simple  or  compound.  By  a compound  event  is  meant  one  of 
which  the  expression  in  language,  or  the  conception  in  thought, 
depends  upon  the  expression  or  the  conception1  of  other  events, 
which,  in  relation  to  it,  may  be  regarded  as  simple  events.  To 
say  “ it  rains,”  or  to  say  “ it  thunders,”  is  to  express  the  occur- 
rence of  a simple  event;  but  to  say  “it  rains  and  thunders,”  or 
to  say  “ it  either  rains  or  thunders,”  is  to  express  that  of  a com1 
pound  event.  For  the  expression  of  that  event  depends  upon 
the  elementary  expressions,  “it  rains,”  “ it  thunders.”  The  cri- 
terion of  simple  events  is  not,  therefore,  any  supposed  simplicity 
in  their  nature.  It  is  founded  solely  on  the  mode  of  their  ex- 
pression in  language  or  conception  in  thought. 

14.  Now  one  general  problem,  which  the  existing  theory  of 
Probabilities  enables  us  to  solve,  is  the  following,  viz. : — Given 


15 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

the  probabilities  of  any  simple  events  : required  the  probability  of 
a given  compound  event,  i.  e.  of  an  event  compounded  in  a given 
manner  out  of  the  given  simple  events.  The  problem  can  also 
be  solved  when  the  compound  event,  whose  probability  is  re- 
quired, is  subjected  to  given  conditions,  i.  e.  to  conditions  de- 
pendent also  in  a given  manner  on  the  given  simple  events. 
Beside  this  general  problem,  there  exist  also  particular  problems 
of  which  the  principle  of  solution  is  known.  Various  questions 
relating  to  causes  and  effects  can  be  solved  by  known  methods 
under  the  particular  hypothesis  that  the  causes  are  mutually  ex- 
clusive, but  apparently  not  otherwise.  Beyond  this  it  is  not 
clear  that  any  advance  has  been  made  toward  the  solution  of 
what  may  be  regarded  as  the  general  problem  of  the  science,  viz. : 
Given  the  probabilities  of  any  events,  simple  or  compound,  con- 
ditioned or  unconditioned : required  the  probability  of  any  other 
event  equally  arbitrary  in  expression  and  conception.  In  the 
statement  of  this  question  it  is  not  even  postulated  that  the 
events  whose  probabilities  are  given,  and  the  one  whose  proba- 
bility is  sought,  should  involve  some  common  elements,  because 
it  is  the  office  of  a method  to  determine  whether  the  data  of  a 
problem  are  sufficient  for  the  end  in  view,  and  to  indicate,  when 
they  are  not  so,  wherein  the  deficiency  consists. 

This  problem,  in  the  most  unrestricted  form  of  its  statement, 
is  resolvable  by  the  method  of  the  present  treatise ; or,  to  speak 
more  precisely,  its  theoretical  solution  is  completely  given,  and 
its  practical  solution  is  brought  to  depend  only  upon  processes 
purely  mathematical,  such  as  the  resolution  and  analysis  of  equa- 
tions. The  order  and  character  of  the  general  solution  may  be 
thus  described. 

15.  In  the  first  place  it  is  always  possible,  by  the  preliminary 
method  of  the  Calculus  of  Logic,  to  express  the  event  whose 
probability  is  sought  as  a logical  function  of  the  events  whose 
probabilities  are  given.  The  result  is  of  the  following  character  : 
Suppose  that  X represents  the  event  whose  probability  is  sought, 
A , B,  C,  &c.  the  events  whose  probabilities  are  given,  those 
events  being  either  simple  or  compound.  Then  the  ivhole  rela- 
tion of  the  event  X to  the  events  A,  B,  C,  &c.  is  deduced  in  the 
form  of  what  mathematicians  term  a development,  consisting,  in 


16 


NATURE  AND  DESIGN  OF  THIS  WORK-  [CHAP.  I. 

the  most  general  case,  of  four  distinct  classes  of  terms.  By  the 
first  class  are  expressed  those  combinations  of  the  events  A,  B,  C, 
which  both  necessarily  accompany  and  necessarily  indicate  the 
occurrence  of  the  event  X ; by  the  second  class,  those  combina- 
tions which  necessarily  accompany,  but  do  not  necessarily  imply, 
the  occurrence  of  the  event  X ; by  the  third  class,  those  combi- 
nations whose  occurrence  in  connexion  with  the  event  X is  im- 
possible, but  not  otherwise  impossible  ; by  the  fourth  class, 
those  combinations  whose  occurrence  is  impossible  under  any  cir- 
cumstances. I shall  not  dwell  upon  this  statement  of  the  result 
of  the  logical  analysis  of  the  problem,  further  than  to  remark 
that  the  elements  which  it  presents  are  precisely  those  by  which 
the  expectation  of  the  event  X,  as  dependent  upon  our  know- 
ledge of  the  events  A,  B,  C,  is,  or  alone  can  be,  affected.  General 
reasoning  would  verify  this  conclusion ; but  general  reasoning 
would  not  usually  avail  to  disentangle  the  complicated  web  of 
events  and  circumstances  from  which  the  solution  above  de- 
scribed must  be  evolved.  The  attainment  of  this  object  consti- 
tutes the  first  step  towards  the  complete  solution  of  the  question 
proposed.  It  is  to  be  noted  that  thus  far  the  process  of  solution 
is  logical,  i.  e.  conducted  by  symbols  of  logical  significance,  and 
resulting  in  an  equation  interpretable  into  a proposition.  Let  this 
result  be  termed  the  final  logical  equation. 

The  second  step  of  the  process  deserves  attentive  remark. 
From  the  final  logical  equation  to  which  the  previous  step  has 
conducted  us,  are  deduced,  by  inspection,  a series  of  algebraic 
equations  implicitly  involving  the  complete  solution  of  the  pro- 
blem proposed.  Of  the  mode  in  which  this  transition  is  effected 
let  it  suffice  to  say,  that  there  exists  a definite  relation  between 
the  laws  by  which  the  probabilities  of  events  are  expressed  as 
algebraic  functions  of  the  probabilities  of  other  events  upon  which 
they  depend,  and  the  laws  by  which  the  logical  connexion  of 
the  events  is  itself  expressed.  This  relation,  like  the  other  co- 
incidences of  formal  law  which  have  been  referred  to,  is  not 
founded  upon  hypothesis,  but  is  made  known  to  us  by  observation 
(1. 4),  and  reflection.  If,  however,  its  reality  were  assumed  a priori 
as  the  basis  of  the  very  definition  of  Probability,  strict  deduction 
■would  thence  lead  us  to  the  received  numerical  definition  as  a 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK.  IT 

necessary  consequence.  The  Theory  of  Probabilities  stands,  as 
it  has  already  been  remarked  (1. 12),  in  equally  close  relation  to 
Logic  and  to  Arithmetic ; and  it  is  indifferent,  so  far  as  results 
are  concerned,  whether  we  regard  it  as  springing  out  of  the  lat- 
ter of  these  sciences,  or  as  founded  in  the  mutual  relations  which 
connect  the  two  together. 

16.  There  are  some  circumstances,  interesting  perhaps  to  the 
mathematician,  attending  the  general  solutions  deduced  by  the 
above  method,  which  it  may  be  desirable  to  notice. 

1st.  As  the  method  is  independent  of  the  number  and  the 
nature  of  the  data,  it  continues  to  be  applicable  when  the  latter 
are  insufficient  to  render  determinate  the  value  sought.  When 
such  is  the  case,  the  final  expression  of  the  solution  will  contain 
terms  with  arbitrary  constant  coefficients.  To  such  terms  there 
will  correspond  terms  in  the  final  logical  equation  (I.  15),  the 
interpretation  of  which  will  inform  us  what  new  data  are  re- 
quisite in  order  to  determine  the  values  of  those  constants,  and 
thus  render  the  numerical  solution  complete.  If  such  data  are 
not  to  be  obtained,  we  can  still,  by  giving  to  the  constants  their 
limiting  values  0 and  1,  determine  the  limits  within  which  the 
probability  sought  must  lie  independently  of  all  further  expe- 
rience. When  the  event  whose  probability  is  sought  is  quite  in- 
dependent of  those  whose  probabilities  are  given,  the  limits  thus 
obtained  for  its  value  will  be  0 and  1 , as  it  is  evident  that  they 
ought  to  be,  and  the  interpretation  of  the  constants  will  only 
lead  to  a re-statement  of  the  original  problem. 

2ndly.  The  expression  of  the  final  solution  will  in  all  cases 
involve  a particular  element  of  quantity,  determinable  by  the  so- 
lution of  an  algebraic  equation.  Now  when  that  equation  is  of 
an  elevated  degree,  a difficulty  may  seem  to  arise  as  to  the  se- 
lection of  the  proper  root.  There  are,  indeed,  cases  in  which 
both  the  elements  given  and  the  element  sought  are  so  obviously 
restricted  by  necessary  conditions  that  no  choice  remains.  But 
in  complex  instances  the  discovery  of  such  conditions,  by  un- 
assisted force  of  reasoning,  would  be  hopeless.  A distinct  me- 
thod is  requisite  for  this  end, — a method  which  might  not 
inappropriately  be  termed  the  Calculus  of  Statistical  Conditions. 
Into  the  nature  of  this  method  I shall  not  here  further  enter 


18 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

than  to  say,  that,  like  the  previous  method,  it  is  based  upon  the 
employment  of  the  “ final  logical  equation,”  and  that  it  definitely 
assigns,  1st,  the  conditions  which  must  be  fulfilled  among  the 
numerical  elements  of  the  data,  in  order  that  the  problem  may 
be  real,  i.  e.  dei'ived  from  a possible  experience  ; 2ndly,  the  nu- 
merical limits,  within  which  the  probability  sought  must  have 
been  confined,  if,  instead  of  being  determined  by  theory,  it  had 
been  deduced  directly  by  observation  from  the  same  system  of 
phenomena  from  which  the  data  were  derived.  It  is  clear  that 
these  limits  will  be  actual  limits  of  the  probability  sought. 
Now,  on  supposing  the  data  subject  to  the  conditions  above  as- 
signed to  them,  it  appears  in  every  instance  which  I have  exa- 
mined that  there  exists  one  root,  and  only  one  root,  of  the  final 
algebraic  equation  which  is  subject  to  the  required  limitations. 
Every  source  of  ambiguity  is  thus  removed.  It  would  even  seem 
that  new  truths  relating  to  the  theory  of  algebraic  equations 
are  thus  incidentally  brought  to  light.  It  is  remarkable  that 
the  special  element  of  quantity,  to  which  the  previous  discussion 
relates,  depends  only  upon  the  data,  and  not  at  all  upon  the 
qucesitum  of  the  problem  proposed.  Hence  the  solution  of  each 
particular  problem  unties  the  knot  of  difficulty  for  a system  of 
problems,  viz.,  for  that  system  of  problems  which  is  marked  by 
the  possession  of  common  data,  independently  of  the  nature  of 
their  quoesita.  This  circumstance  is  important  whenever  from  a 
particular  system  of  data  it  is  required  to  deduce  a series  of  con- 
nected conclusions.  And  it  further  gives  to  the  solutions  of 
particular  problems  that  character  of  relationship,  derived  from 
their  dependence  upon  a central  and  fundamental  unity,  which 
not  unfrequently  marks  the  application  of  general  methods. 

17.  But  though  the  above  considerations,  with  others  of  a 
like  nature,  justify  the  assertion  that  the  method  of  this  treatise, 
for  the  solution  of  questions  in  the  theory  of  Probabilities,  is  a 
general  method,  it  does  not  thence  follow  that  we  are  relieved  in 
all  cases  from  the  necessity  of  recourse  to  hypothetical  grounds. 
It  has  been  observed  that  a solution  may  consist  entirely  of  terms 
affected  by  arbitrary  constant  coefficients, — may,  in  fact,  be 
wholly  indefinite.  The  application  of  the  method  of  this  work  to 
some  of  the  most  important  questions  within  its  range  would — 


19 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

were  the  data  of  experience  alone  employed — present  results  of 
this  character.  To  obtain  a definite  solution  it  is  necessary,  in 
such  cases,  to  have  recourse  to  hypotheses  possessing  more  or  less 
of  independent  probability,  but  incapable  of  exact  verification. 
Generally  speaking,  such  hypotheses  will  differ  from  the  imme- 
diate results  of  experience  in  partaking  of  a logical  rather  than  of  a 
numerical  character ; in  prescribing  the  conditions  under  which 
phenomena  occur,  rather  than  assigning  the  relative  frequency 
of  their  occurrence.  This  circumstance  is,  however,  unimportant. 
Whatever  their  nature  may  be,  the  hypotheses  assumed  must 
thenceforth  be  regarded  as  belonging  to  the  actual  data,  although 
tending,  as  is  obvious,  to  give  to  the  solution  itself  somewhat  of 
a hypothetical  character.  With  this  understanding  as  to  the 
possible  sources  of  the  data  actually  employed,  the  method  is 
perfectly  general,  but  for  the  correctness  of  the  hypothetical  ele- 
ments introduced  it  is  of  course  no  more  responsible  than  for  the 
correctness  of  the  numerical  data  derived  from  experience. 

In  illustration  of  these  remarks  we  may  observe  that  the 
theory  of  the  reduction  of  astronomical  observations*  rests,  in 
part,  upon  hypothetical  grounds.  It  assumes  certain  positions 
as  to  the  nature  of  error,  the  equal  probabilities  of  its  occurrence 
in  the  form  of  excess  or  defect,  &c.,  without  which  it  would  be 
impossible  to  obtain  any  definite  conclusions  from  a system  of 
conflicting  observations.  But  granting  such  positions  as  the 
above,  the  residue  of  the  investigation  falls  strictly  within  the 
province  of  the  theory  of  Probabilities.  Similar  observations 
apply  to  the  important  problem  which  proposes  to  deduce  from 
the  records  of  the  majorities  of  a deliberative  assembly  the  mean 
probability  of  correct  judgment  in  one  of  its  members.  If  the 
method  of  this  treatise  be  applied  to  the  mere  numerical  data, 
the  solution  obtained  is  of  that  wholly  indefinite  kind  above  de- 
scribed. And  to  show  in  a more  eminent  degree  the  insufficiency 
of  those  data  by  themselves,  the  interpretation  of  the  arbitrary 
constants  (I.  16)  which  appear  in  the  solution,  merely  produces 

* The  author  designs  to  treat  this  subject  either  in  a separate  work  or  in  a 
future  Appendix.  In  the  present  treatise  he  avoids  the  use  of  the  integral 
calculus. 


20 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

a re-statement  of  the  original  problem.  Admitting,  however, 
the  hypothesis  of  the  independent  formation  of  opinion  in  the 
individual  mind,  either  absolutely,  as  in  the  speculations  of 
Laplace  and  Poisson,  or  under  limitations  imposed  by  the  actual 
data,  as  will  be  seen  in  this  treatise,  Chap,  xxi.,  the  problem  as- 
sumes a far  more  definite  character.  It  will  be  manifest  that  the 
ulterior  value  of  the  theory  of  Probabilities  must  depend  very 
much  upon  the  correct  formation  of  such  mediate  hypotheses, 
where  the  purely  experimental  data  are  insufficient  for  definite 
solution,  and  where  that  further  experience  indicated  by  the  in- 
terpretation of  the  final  logical  equation  is  unattainable.  Upon 
the  other  hand,  an  undue  readiness  to  form  hypotheses  in  sub- 
jects which  from  their  very  nature  are  placed  beyond  human 
ken,  must  re-act  upon  the  credit  of  the  theory  of  Probabilities, 
and  tend  to  throw  doubt  in  the  general  mind  over  its  most  legi- 
timate conclusions. 

18.  It  would,  perhaps,  be  premature  to  speculate  here  upon 
the  question  whether  the  methods  of  abstract  science  are  likely  at 
any  future  day  to  render  service  in  the  investigation  of  social 
problems  at  all  commensurate  with  those  which  they  have  ren- 
dered in  various  departments  of  physical  inquiry.  An  attempt 
to  resolve  this  question  upon  pure  a priori  grounds  of  reasoning 
would  be  very  likely  to  mislead  us.  For  example,  the  conside- 
ration of  human  free-agency  would  seem  at  first  sight  to  preclude 
the  idea  that  the  movements  of  the  social  system  should  ever  ma- 
nifest that  character  of  orderly  evolution  which  we  are  prepared 
to  expect  under  the  reign  of  a physical  necessity.  Yet  already 
do  the  researches  of  the  statist  reveal  to  us  facts  at  variance  with 
such  an  anticipation.  Thus  the  records  of  crime  and  pauperism 
present  a degree  of  regularity  unknown  in  regions  in  which  the 
disturbing  influence  of  human  wants  and  passions  is  unfelt.  On 
the  other  hand,  the  distemperature  of  seasons,  the  eruption  of 
volcanoes,  the  spread  of  blight  in  the  vegetable,  or  of  epidemic 
maladies  in  the  animal  kingdom,  things  apparently  or  chiefly  the 
product  of  natural  causes,  refuse  to  be  submitted  to  regular  and 
apprehensible  laws.  “ Fickle  as  the  wind,”  is  a proverbial  ex- 
pression. Reflection  upon  these  points  teaches  us  in  some  degree 
to  correct  our  earlier  judgments.  We  learn  that  we  are  not  to 


21 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK. 

expect,  under  the  dominion  of  necessity,  an  order  perceptible  to 
human  observation,  unless  the  play  of  its  producing  causes  is 
sufficiently  simple ; nor,  on  the  other  hand,  to  deem  that  free 
agency  in  the  individual  is  inconsistent  with  regularity  in  the 
motions  of  the  system  of  which  he  forms  a component  unit. 
Human  freedom  stands  out  as  an  apparent  fact  of  our  conscious- 
ness, while  it  is  also,  I conceive,  a highly  probable  deduction  of 
analogy  (Chap,  xxn.)  from  the  nature  of  that  portion  of  the 
mind  whose  scientific  constitution  we  are  able  to  investigate. 
But  whether  accepted  as  a fact  reposing  on  consciousness,  or  as 
a conclusion  sanctioned  by  the  reason,  it  must  be  so  interpreted 
as  not  to  conflict  with  an  established  result  of  observation,  viz. : 
that  phenomena,  in  the  production  of  which  large  masses  of  men 
are  concerned,  do  actually  exhibit  a very  remarkable  degree  of 
regularity,  enabling  us  to  collect  in  each  succeeding  age  the  ele- 
ments upon  which  the  estimate  of  its  state  and  progress,  so  far 
as  manifested  in  outward  results,  must  depend.  There  is  thus  no 
sound  objection  a priori  against  the  possibility  of  that  species  of 
data  which  is  requisite  for  the  experimental  foundation  of  a 
science  of  social  statistics.  Again,  whatever  other  object  this 
treatise  may  accomplish,  it  is  presumed  that  it  will  leave  no 
doubt  as  to  the  existence  of  a system  of  abstract  principles  and  of 
methods  founded  upon  those  principles,  by  which  any  collective 
body  of  social  data  may  be  made  to  yield,  in  an  explicit  form, 
whatever  information  they  implicitly  involve.  There  may,  where 
the  data  are  exceedingly  complex,  be  very  great  difficulty  in  ob- 
taining this  information, — difficulty  due  not  to  any  imperfection 
of  the  theory,  but  to  the  laborious  character  of  the  analytical 
processes  to  which  it  points.  It  is  quite  conceivable  that  in  many 
instances  that  difficulty  may  be  such  as  only  united  effort  could 
overcome.  But  that  we  possess  theoretically  in  all  cases,  and 
practically,  so  far  as  the  requisite  labour  of  calculation  may  be 
supplied,  the  means  of  evolving  from  statistical  records  the  seeds 
of  general  truths  which  lie  buried  amid  the  mass  of  figures,  is  a 
position  which  may,  I conceive,  with  perfect  safety  be  affirmed. 

19.  But  beyond  these  general  positions  I do  not  venture  to 
speak  in  terms  of  assurance.  Whether  the  results  which  might 
be  expected  from  the  application  of  scientific  methods  to  statis- 


22 


NATURE  AND  DESIGN  OF  THIS  WORK.  [CHAP.  I. 

tical  records,  over  and  above  those  the  discovery  of  which  re- 
quires no  such  aid,  would  so  far  compensate  for  the  labour  in- 
volved as  to  render  it  worth  while  to  institute  such  investigations 
upon  a proper  scale  of  magnitude,  is  a point  which  could,  per- 
haps, only  be  determined  by  experience.  It  is  to  be  desired, 
and  it  might  without  great  presumption  be  expected,  that  in 
this,  as  in  other  instances,  the  abstract  doctrines  of  science  should 
minister  to  more  than  intellectual  gratification.  Nor,  viewing 
the  apparent  order  in  which  the  sciences  have  been  evolved,  and 
have  successively  contributed  their  aid  to  the  service  of  mankind, 
does  it  seem  very  improbable  that  a day  may  arrive  in  which  si- 
milar aid  may  accrue  from  departments  of  the  field  of  knowledge 
yet  more  intimately  allied  with  the  elements  of  human  welfare. 
Let  the  speculations  of  this  treatise,  however,  rest  at  present 
simply  upon  them  claim  to  be  regarded,  as  true. 

20.  I design,  in  the  last  place,  to  endeavour  to  educe  from 
the  scientific  results  of  the  previous  inquiries  some  general  inti- 
mations respecting  the  nature  and  constitution  of  the  human 
mind.  Into  the  grounds  of  the  possibility  of  this  species  of  in- 
ference it  is  not  necessary  to  enter  here.  One  or  two  general 
observations  may  serve  to  indicate  the  track  which  I shall  endea- 
vour to  follow.  It  cannot  but  be  admitted  that  our  views  of 
the  science  of  Logic  must  materially  influence,  perhaps  mainly 
determine,  our  opinions  upon  the  nature  of  the  intellectual  facul- 
ties. For  example,  the  question  whether  reasoning  consists 
merely  in  the  application  of  certain  first  or  necessary  truths, 
with  which  the  mind  has  been  originally  imprinted,  or  whether 
the  mind  is  itself  a seat  of  law,  whose  operation  is  as  manifest 
and  as  conclusive  in  the  particular  as  in  the  general  formula,  or 
whether,  as  some  not  undistinguished  writers  seem  to  maintain, 
all  reasoning  is  of  particulars ; this  question,  I say,  is  one  which 
not  merely  affects  the  science  of  Logic,  but  also  concerns  the  for- 
mation of  just  views  of  the  constitution  of  the  intellectual  facul- 
ties. Again,  if  it  is  concluded  that  the  mind  is  by  original 
constitution  a seat  of  law,  the  question  of  the  nature  of  its  sub- 
jection to  this  law, — whether,  for  instance,  it  is  an  obedience 
founded  upon  necessity,  like  that  which  sustains  the  revolutions 
of  the  heavens,  and  preserves  the  order  of  Nature, — or  whether 


CHAP.  I.]  NATURE  AND  DESIGN  OF  THIS  WORK.  23 

it  is  a subjection  of  some  quite  distinct  kind,  is  also  a matter  of 
deep  speculative  interest.  Further,  if  the  mind  is  truly  deter- 
mined to  be  a subject  of  law,  and  if  its  laws  also  are  truly  assigned, 
the  question  of  their  probable  or  necessary  influence  upon  the 
course  of  human  thought  in  different  ages  is  one  invested  with 
great  importance,  and  well  deserving  a patient  investigation,  as 
matter  both  of  philosophy  and  of  history.  These  and  other 
questions  I propose,  however  imperfectly,  to  discuss  in  the  con- 
cluding portion  of  the  present  work.  They  belong,  perhaps,  to 
the  domain  of  probable  or  conjectural,  rather  than  to  that  of  po- 
sitive, knowledge.  But  it  may  happen  that  where  there  is  not 
sufficient  warrant  for  the  certainties  of  science,  there  may  be 
grounds  of  analogy  adequate  for  the  suggestion  of  highly  pro- 
bable opinions.  It  has  seemed  to  me  better  that  this  discussion 
should  be  entirely  reserved  for  the  sequel  of  the  main  business  of 
this  treatise, — which  is  the  investigation  of  scientific  truths  and 
laws.  Experience  sufficiently  instructs  us  that  the  proper  order 
of  advancement  in  all  inquiries  after  truth  is  to  proceed  from  the 
known  to  the  unknown.  There  are  parts,  even  of  the  philosophy 
and  constitution  of  the  human  mind,  which  have  been  placed 
fully  within  the  reach  of  our  investigation.  To  make  a due  ac- 
quaintance with  those  portions  of  our  nature  the  basis  of  all  en- 
deavours to  penetrate  amid  the  shadows  and  uncertainties  of  that 
conjectural  realm  which  lies  beyond  and  above  them,  is  the 
course  most  accordant  with  the  limitations  of  our  present  con- 
dition. 


24 


SIGNS  AND  THEIR  LAWS. 


[CHAP.  II. 


CHAPTER  II. 

OF  SIGNS  IN  GENERAL,  AND  OF  THE  SIGNS  APPROPRIATE  TO  THE 
SCIENCE  OF  LOGIC  IN  PARTICULAR  ; ALSO  OF  THE  LAWS  TO  WHICH 

THAT  CLASS  OF  SIGNS  ARE  SUBJECT. 

1 • nPHAT  Language  is  an  instrument  of  human  reason,  and 
not  merely  a medium  for  the  expression  of  thought,  is  a 
truth  generally  admitted.  It  is  proposed  in  this  chapter  to  in- 
quire what  it  is  that  renders  Language  thus  subservient  to  the 
most  important  of  our  intellectual  faculties.  In  the  various 
steps  of  this  inquiry  we  shall  be  led  to  consider  the  constitution 
of  Language,  considered  as  a system  adapted  to  an  end  or  pur- 
pose ; to  investigate  its  elements ; to  seek  to  determine  their  mu- 
tual relation  and  dependence ; and  to  inquire  in  what  manner  they 
contribute  to  the  attainment  of  the  end  to  which,  as  co-ordinate 
parts  of  a system,  they  have  respect. 

In  proceeding  to  these  inquiries,  it  will  not  be  necessary  to 
enter  into  the  discussion  of  that  famous  question  of  the  schools, 
whether  Language  is  to  be  regarded  as  an  essential  instrument 
of  reasoning,  or  whether,  on  the  other  hand,  it  is  possible  for  us 
to  reason  without  its  aid.  I suppose  this  question  to  be  beside 
the  design  of  the  present  treatise,  for  the  following  reason,  viz., 
that  it  is  the  business  of  Science  to  investigate  laws ; and  that, 
whether  we  regard  signs  as  the  representatives  of  things  and  of 
their  relations,  or  as  the  representatives  of  the  conceptions  and 
operations  of  the  human  intellect,  in  studying  the  laws  of  signs, 
we  are  in  effect  studying  the  manifested  laws  of  reasoning.  If 
there  exists  a difference  between  the  two  inquiries,  it  is  one  which 
does  not  affect  the  scientific  expressions  of  formal  law,  which  are 
the  object  of  investigation  in  the  present  stage  of  this  work,  but 
relates  only  to  the  mode  in  which  those  results  are  presented  to 
the  mental  regard.  For  though  in  investigating  the  laws  of  signs, 
a posteriori,  the  immediate  subject  of  examination  is  Language, 
with  the  rules  which  govern  its  use ; while  in  making  the  internal 


SIGNS  AND  THEIR  LAWS. 


25 


CHAP.  II.] 


processes  of  thought  the  direct  object  of  inquiry,  we  appeal  in  a 
more  immediate  way  to  our  personal  consciousness, — it  will  be 
found  that  in  both  cases  the  results  obtained  are  formally  equi- 
valent. Nor  could  we  easily  conceive,  that  the  unnumbered 
tongues  and  dialects  of  the  earth  should  have  preserved  through 
a long  succession  of  ages  so  much  that  is  common  and  universal, 
were  we  not  assured  of  the  existence  of  some  deep  foundation  of 
their  agreement  in  the  laws  of  the  mind  itself. 

2.  The  elements  of  which  all  language  consists  are  signs  or 
symbols.  Words  are  signs.  Sometimes  they  are  said  to  repre- 
sent things  ; sometimes  the  operations  by  which  the  mind  com- 
bines together  the  simple  notions  of  things  into  complex  concep- 
tions ; sometimes  they  express  the  relations  of  action,  passion,  or 
mere  quality,  which  we  perceive  to  exist  among  the  objects  of  our 
experience ; sometimes  the  emotions  of  the  perceiving  mind.  But 
words,  although  in  this  and  in  other  ways  they  fulfil  the  office  of 
signs,  or  representative  symbols,  are  not  the  only  signs  which  we 
are  capable  of  employing.  Arbitrary  marks,  which  speak  only  to 
the  eye,  and  arbitrary  sounds  or  actions,  which  address  themselves 
to  some  other  sense,  are  equally  of  the  nature  of  signs,  provided 
that  their  representative  office  is  defined  and  understood.  In  the 
mathematical  sciences,  letters,  and  the  symbols  +,  -,  =,  &c.,  are 
used  as  signs,  although  the  term  “ sign”  is  applied  to  the  latter 
class  of  symbols,  which  represent  operations  or  relations,  rather 
than  to  the  former,  which  represent  the  elements  of  number  and 
quantity.  As  the  real  import  of  a sign  does  not  in  any  way  de- 
pend upon  its  particular  form  or  expression,  so  neither  do  the 
laws  which  determine  its  use.  In  the  present  treatise,  however, 
it  is  with  written  signs  that  we  have  to  do,  and  it  is  with  reference 
to  these  exclusively  that  the  term  “ sign”  will  be  employed.  The 
essential  properties  of  signs  are  enumerated  in  the  following  de- 
finition. 

Definition. — A sign  is  an  arbitrary  mark,  having  a fixed  in- 
terpretation, and  susceptible  of  combination  with  other  signs  in 
subjection  to  fixed  laws  dependent  upon  their  mutual  interpre- 
tation. 

3.  Let  us  consider  the  particulars  involved  in  the  above  de- 
finition separately. 


26 


SIGNS  AND  THEIR  LAWS. 


[CHAP.  II. 

(1.)  In  the  first  place,  a sign  is  an  arbitrary  mark.  It  is 
clearly  indifferent  what  particular  word  or  token  we  associate 
with  a given  idea,  provided  that  the  association  once  made  is 
permanent.  The  Romans  expressed  by  the  word  “ civitas”  what 
we  designate  by  the  word  “ state.”  But  both  they  and  we 
might  equally  well  have  employed  any  other  word  to  represent 
the  same  conception.  Nothing,  indeed,  in  the  nature  of  Language 
would  prevent  us  from  using  a mere  letter  in  the  same  sense. 
Were  this  done,  the  laws  according  to  which  that  letter  would 
require  to  be  used  would  be  essentially  the  same  with  the  laws 
which  govern  the  use  of  “civitas”  in  the  Latin,  and  of  “state” 
in  the  English  language,  so  far  at  least  as  the  use  of  those  words 
is  regulated  by  any  general  principles  common  to  all  languages 
alike. 

(2.)  In  the  second  place,  it  is  necessary  that  each  sign  should 
possess,  within  the  limits  of  the  same  discourse  or  process  of 
reasoning,  a fixed  interpretation.  The  necessity  of  this  condi- 
tion is  obvious,  and  seems  to  be  founded  in  the  very  nature  of  the 
subject.  There  exists,  however,  a dispute  as  to  the  precise  nature 
of  the  representative  office  of  words  or  symbols  used  as  names  in 
t he  processes  of  reasoning.  By  some  it  is  maintained,  that  they 
represent  the  conceptions  of  the  mind  alone ; by  others,  that  they 
represent  things.  The  question  is  not  of  great  importance  here, 
as  its  decision  cannot  affect  the  laws  according  to  which  signs 
are  employed.  I apprehend,  however,  that  the  general  answer 
to  this  and  such  like  questions  is,  that  in  the  processes  of  reason- 
ing, signs  stand  in  the  place  and  fulfil  the  office  of  the  concep- 
tions and  operations  of  the  mind ; but  that  as  those  conceptions 
and  operations  represent  things,  and  the  connexions  and  relations 
of  things,  so  signs  represent  things  with  their  connexions  and  re- 
lations ; and  lastly,  that  as  signs  stand  in  the  place  of  the  con- 
ceptions and  operations  of  the  mind,  they  are  subject  to  the  laws 
of  those  conceptions  and  operations.  This  view  will  be  more 
fully  elucidated  in  the  next  chapter ; but  it  here  serves  to  explain 
the  third  of  those  particulars  involved  in  the  definition  of  a sign, 
viz.,  its  subjection  to  fixed  laws  of  combination  depending  upon 
the  nature  of  its  interpretation. 

4.  The  analysis  and  classification  of  those  signs  by  which  the 


CHAP.  II.]  SIGNS  AND  THEIR  LAWS.  27 

operations  of  reasoning  are  conducted  will  be  considered  in  the 
following  Proposition : 

Proposition  I. 

All  the  operations  of  Language,  as  an  instrument  of  reasoning, 
may  be  conducted  by  a system  of  signs  composed  of  the  following  ele- 
ments, viz.  : 

1st.  Literal  symbols,  as  x , y,  Sfc.,  representing  things  as  subjects 
of  our  conceptions. 

2nd.  Signs  of  operation,  as  + ,-,x,  standing  for  those  operations 
of  the  mind  by  which  the  conceptions  of  things  are  combined  or  re- 
solved so  as  to  form  new  conceptions  involving  the  same  elements. 

3rd.  The  sign  of  identity,  =. 

And  these  symbols  of  Logic  are  in  their  use  subject  to  definite 
laws,  partly  agreeing  with  and  partly  differing  from  the  laws  of  the 
corresponding  symbols  in  the  science  of  Algebra. 

Let  it  be  assumed  as  a criterion  of  the  true  elements  of  ra- 
tional discourse,  that  they  should  be  susceptible  of  combination 
in  the  simplest  forms  and  by  the  simplest  laws,  and  thus  com- 
bining should  generate  all  other  known  and  conceivable  forms  of 
language ; and  adopting  this  principle,  let  the  following  classifi- 
cation be  considered. 

class  i. 

5.  Appellative  or  descriptive  signs,  expressing  either  the  name 
of  a thing,  or  some  quality  or  circumstance  belonging  to  it. 

To  this  class  we  may  obviously  refer  the  substantive  proper 
or  common,  and  the  adjective.  These  may  indeed  be  regarded  as 
differing  only  in  this  respect,  that  the  former  expresses  the  sub- 
stantive existence  of  the  individual  thing  or  things  to  which  it 
refers ; the  latter  implies  that  existence.  If  we  attach  to  the 
adjective  the  universally  understood  subject  “ being”  or  “ thing,” 
it  becomes  virtually  a substantive,  and  may  for  all  the  essential 
purposes  of  reasoning  be  replaced  by  the  substantive.  Whether 
or  not,  in  every  particular  of  the  mental  regard,  it  is  the  same 
thing  to  say,  “Water  is  a fluid  thing,”  as  to  say,  “Water  is 
fluid it  is  at  least  equivalent  in  the  expression  of  the  processes 
of  reasoning. 


28 


SIGNS  AND  THEIR  LAWS. 


[CHAP.  II. 


It  is  clear  also,  that  to  the  above  class  we  must  refer  any  smn 
which  may  conventionally  be  used  to  express  some  circumstance 
or  relation,  the  detailed  exposition  of  which  would  involve  the 
use  of  many  signs.  The  epithets  of  poetic  diction  are  very  fre- 
quently of  this  kind.  They  are  usually  compounded  adjectives, 
singly  fulfilling  the  office  of  a many- worded  description.  Homer’s 
“ deep-eddying  ocean”  embodies  a virtual  description  in  the  single 
word  j3a0i/S/vr}c.  And  conventionally  any  other  description  ad- 
dressed either  to  the  imagination  or  to  the  intellect  might  equally 
be  represented  by  a single  sign,  the  use  of  which  would  in  all  es- 
sential points  be  subject  to  the  same  laws  as  the  use  of  the  ad- 
jective “ good”  or  “ great.”  Combined  with  the  subject  “ thing,” 
such  a sign  would  virtually  become  a substantive ; and  by  a single 
substantive  the  combined  meaning  both  of  thing  and  quality 
might  be  expressed. 

6.  Now,  as  it  has  been  defined  that  a sign  is  an  arbitrary 
mark,  it  is  permissible  to  replace  all  signs  of  the  species  above 
described  by  letters.  Let  us  then  agree  to  represent  the  class  of 
individuals  to  which  a particular  name  or  description  is  appli- 
cable, by  a single  letter,  as  x.  If  the  name  is  “ men,”  for  instance, 
let  x represent  “all  men,”  or  the  class  “men.”  By  a class  is 
usually  meant  a collection  of  individuals,  to  each  of  which  a 
particular  name  or  description  may  be  applied ; but  in  this  work 
the  meaning  of  the  term  will  be  extended  so  as  to  include  the 
case  in  which  but  a single  individual  exists,  answering  to  the 
required  name  or  description,  as  well  as  the  cases  denoted  by 
the  terms  “ nothing”  and  “ universe,”  which  as  “ classes” 
should  be  understood  to  comprise  respectively  “ no  beings,” 
“ all  beings.”  Again,  if  an  adjective,  as  “good,”  is  employed 
as  a term  of  description,  let  us  represent  by  a letter,  as  y,  all 
things  to  which  the  description  “ good”  is  applicable,  i.  e.  “ all 
good  things,”  or  the  class  “good  things.”  Let  it  further  be 
agreed,  that  by  the  combination  xy  shall  be  represented  that 
class  of  things  to  which  the  names  or  descriptions  represented  by 
x and  y are  simultaneously  applicable.  Thus,  if  x alone  stands 
for  “ white  things,”  and  y for  “ sheep,”  let  xy  stand  for  “ white 
sheep ;”  and  in  like  manner,  if  z stand  for  “ horned  things,”  and 
x and  y retain  their  previous  interpretations,  let  zxy  represent 


SIGNS  AND  THEIR  LAWS. 


29 


CHAP.  II.] 

“ horned  white  sheep,”  i.  e.  that  collection  of  things  to  which 
the  name  “ sheep,”  and  the  descriptions  “ white”  and  “ horned” 
are  together  applicable. 

Let  us  now  consider  the  laws  to  which  the  symbols  x,  y,  &c., 
used  in  the  above  sense,  are  subject. 

7.  First,  it  is  evident,  that  according  to  the  above  combina- 
tions, the  order  in  which  two  symbols  are  written  is  indifferent. 
The  expressions  xy  and  yx  equally  represent  that  class  of  things 
to  the  several  members  of  which  the  names  or  descriptions  x and 
y are  together  applicable.  Hence  we  have, 

xy  = yx.  (1) 

In  the  case  of  x representing  white  things,  and  y sheep,  either 
of  the  members  of  this  equation  will  represent  the  class  of  “ white 
sheep.”  There  may  be  a difference  as  to  the  order  in  which  the 
conception  is  formed,  but  there  is  none  as  to  the  individual  things 
which  are  comprehended  under  it.  In  like  manner,  if  x represent 
“ estuaries,”  and  y “ rivers,”  the  expressions  xy  and  yx  will  in- 
differently represent  “ rivers  that  are  estuaries,”  or  “ estuaries 
that  are  rivers,”  the  combination  in  this  case  being  in  ordinary 
language  that  of  two  substantives,  instead  of  that  of  a substantive 
and  an  adjective  as  in  the  previous  instance.  Let  there  be  a 
third  symbol,  as  z,  representing  that  class  of  things  to  which  the 
term  “ navigable”  is  applicable,  and  any  one  of  the  following 
expressions, 

zxy,  zyx , xyz,  &c., 

will  represent  the  class  of  “ navigable  rivers  that  are  estuaries.” 

If  one  of  the  descriptive  terms  should  have  some  implied  re- 
ference to  another,  it  is  only  necessary  to  include  that  reference 
expressly  in  its  stated  meaning,  in  order  to  render  the  above 
remarks  still  applicable.  Thus,  if  x represent  “wise”  and  y 
“ counsellor,”  we  shall  have  to  define  whether  x implies  wisdom 
in  the  absolute  sense,  or  only  the  wisdom  of  counsel.  With  such 
definition  the  law  xy  - yx  continues  to  be  valid. 

We  are  permitted,  therefore,  to  employ  the  symbols  x,  y,  z,  Sfc.,  in 
the  place  of  the  substantives,  adjectives,  and  descriptive  phrases  subject 
to  the  rule  of  interpretation,  that  any  expression  in  which  several  of 
these  symbols  are  written  together  shall  represent  all  the  objects  or  indi- 


30  SIGNS  AND  THEIR  LAWS.  [CHAP.  II. 

viduals  to  which  their  several  meanings  are  together  applicable,  and 
to  the  law  that  the  order  in  which  the  symbols  succeed  each  other  is 
indifferent. 

As  the  rule  of  interpretation  has  been  sufficiently  exempli- 
fied, I shall  deem  it  unnecessary  always  to  express  the  subject 
“ things”  in  defining  the  interpretation  of  a symbol  used  for  an 
adjective.  When  I say,  let  x represent  45  good,”  it  will  be  un- 
derstood that  x only  represents  44  good”  when  a subject  for  that 
quality  is  supplied  by  another  symbol,  and  that,  used  alone,  its  in- 
terpretation will  be  44  good  things.” 

8.  Concerning  the  law  above  determined,  the  following  ob- 
servations, which  will  also  be  more  or  less  appropriate  to  certain 
other  laws  to  be  deduced  hereafter,  may  be  added. 

First,  I would  remark,  that  this  law  is  a law  of  thought,  and 
not,  properly  speaking,  a law  of  things.  Difference  in  the  order 
of  the  qualities  or  attributes  of  an  object,  apart  from  all  ques- 
tions of  causation,  is  a difference  in  conception  merely.  The  law 
(1)  expresses  as  a general  truth,  that  the  same  thing  may  be  con- 
ceived in  different  ways,  and  states  the  nature  of  that  difference  ; 
and  it  does  no  more  than  this. 

Secondly,  As  a law  of  thought,  it  is  actually  developed  in  a 
law  of  Language,  the  product  and  the  instrument  of  thought. 
Though  the  tendency  of  prose  writing  is  toward  uniformity, 
yet  even  there  the  order  of  sequence  of  adjectives  absolute  in 
their  meaning,  and  applied  to  the  same  subject,  is  indifferent, 
but  poetic  diction  borrows  much  of  its  rich  diversity  from  the 
extension  of  the  same  lawful  freedom  to  the  substantive  also. 
The  language  of  Milton  is  peculiarly  distinguished  by  this  spe- 
cies of  variety.  Not  only  does  the  substantive  often  precede  the 
adjectives  by  which  it  is  qualified,  but  it  is  frequently  placed  in 
their  midst.  In  the  first  few  lines  of  the  invocation  to  Light, 
we  meet  with  such  examples  as  the  following : 

“ Offspring  of  heaven  first-born .” 

“ The  rising  world  of  waters  dark  and  deep.'1'’ 

44  Bright  effluence  of  bright  essence  increate .” 

Now  these  inverted  forms  are  not  simply  the  fruits  of  a poetic 
license.  They  are  the  natural  expressions  of  a freedom  sane- 


CHAP.  II.]  SIGNS  AND  THEIR  LAWS.  31 

tioned  by  the  intimate  laws  of  thought,  but  for  reasons  of  conve- 
nience not  exercised  in  the  ordinary  use  of  language. 

Thirdly,  The  law  expressed  by  (1)  may  be  characterized  by 
saying  that  the  literal  symbols  x,  y,  z,  are  commutative,  like  the 
symbols  of  Algebra . In  saying  this,  it  is  not  affirmed  that  the 
process  of  multiplication  in  Algebra,  of  which  the  fundamental 
law  is  expressed  by  the  equation 

xy  = yx, 

possesses  in  itself  any  analogy  with  that  process  of  logical  com- 
bination which  xy  has  been  made  to  represent  above ; but  only 
that  if  the  arithmetical  and  the  logical  process  are  expressed  in 
the  same  manner,  their  symbolical  expressions  will  be  subject  to 
the  same  formal  law.  The  evidence  of  that  subjection  is  in  the 
two  cases  quite  distinct. 

9.  As  the  combination  of  two  literal  symbols  in  the  form  xy 
expresses  the  whole  of  that  class  of  objects  to  which  the  names 
or  qualities  represented  by  x and  y are  together  applicable,  it 
follows  that  if  the  two  symbols  have  exactly  the  same  significa- 
tion, their  combination  expresses  no  more  than  either  of  the 
symbols  taken  alone  would  do.  In  such  case  we  should  there- 
fore have 

xy  = x. 

As  y is,  however,  supposed  to  have  the  same  meaning  as  x,  we 
may  replace  it  in  the  above  equation  by  x,  and  we  thus  get 

xx  = x. 

Now  in  common  Algebra  the  combination  xx  is  more  briefly  re- 
presented by  x2.  Let  us  adopt  the  same  principle  of  notation 
here ; for  the  mode  of  expressing  a particular  succession  of  mental 
operations  is  a thing  in  itself  quite  as  arbitrary  as  the  mode  of 
expressing  a single  idea  or  operation  (II.  3).  In  accordance  with 
this  notation,  then,  the  above  equation  assumes  the  form 

x2  - x,  (2) 

and  is,  in  fact,  the  expression  of  a second  general  law  of  those 
symbols  by  which  names,  qualities,  or  descriptions,  are  symboli- 
cally represented. 


32  SIGNS  AND  THEIR  LAWS.  [CHAP.  II. 

The  reader  must  bear  in  mind  that  although  the  symbols  x 
and  y in  the  examples  previously  formed  received  significations 
distinct  from  each  other,  nothing  prevents  us  from  attributing  to 
them  precisely  the  same  signification.  It  is  evident  that  the 
more  nearly  their  actual  significations  approach  to  each  other, 
the  more  nearly  does  the  class  of  things  denoted  by  the  combi- 
nation xy  approach  to  identity  with  the  class  denoted  by  x , as 
well  as  with  that  denoted  by  y.  The  case  supposed  in  the  de- 
monstration of  the  equation  (2)  is  that  of  absolute  identity  of 
meaning.  The  law  which  it  expresses  is  practically  exemplified 
in  language.  To  say  “good,  good,”  in  relation  to  any  subject, 
though  a cumbrous  and  useless  pleonasm,  is  the  same  as  to  say 
“good.”  Thus  “good,  good”  men,  is  equivalent  to  “good” 
men.  Such  repetitions  of  words  are  indeed  sometimes  employed 
to  heighten  a quality  or  strengthen  an  affirmation.  But  this 
effect  is  merely  secondary  and  conventional ; it  is  not  founded  in 
the  intrinsic  relations  of  language  and  thought.  Most  of  the 
operations  which  we  observe  in  nature,  or  perform  ourselves,  are 
of  such  a kind  that  their  effect  is  augmented  by  repetition,  and 
this  circumstance  prepares  us  to  expect  the  same  thing  in  lan- 
guage, and  even  to  use  repetition  when  we  design  to  speak  with 
emphasis.  But  neither  in  strict  reasoning  nor  in  exact  discourse 
is  there  any  just  ground  for  such  a practice. 

10.  We  pass  now  to  the  consideration  of  another  class  of  the 
signs  of  speech,  and  of  the  laws  connected  with  their  use. 

CLASS  II. 

11.  Signs  of  those  mental  operations  whereby  we  collect  parts 
into  a whole , or  separate  a whole  into  its  parts. 

We  are  not  only  capable  of  entertaining  the  conceptions  of 
objects,  as  characterized  by  names,  qualities,  or  circumstances, 
applicable  to  each  individual  of  the  group  under  consideration, 
but  also  of  forming  the  aggregate  conception  of  a group,  of  objects 
consisting  of  partial  groups,  each  of  which  is  separately  named 
or  described.  For  this  purpose  we  use  the  conjunctions  “and,” 
“or,”&c.  “ Trees  and  minerals,”  “barren  mountains,  or  fer- 

tile vales,”  are  examples  of  this  kind.  In  strictness,  the  words 


CHAP.  II.] 


SIGNS  AND  THEIR  LAWS. 


33 


“ and,”  “ or,”  interposed  between  the  terms  descriptive  of  two  or 
more  classes  of  objects,  imply  that  those  classes  are  quite  distinct, 
so  that  no  member  of  one  is  found  in  another.  In  this  and  in 
all  other  respects  the  words  “ and”  “ or”  are  analogous  with  the 
sign  + in  algebra,  and  their  laws  are  identical.  Thus  the  ex- 
pression “ men  and  women”  is,  conventional  meanings  set  aside, 
equivalent  with  the  expression  “ women  and  men.”  Let  x repre- 
sent “ men,”  y,  “women  ;”  and  let  + stand  for  “ and ” and  “ or,” 
then  we  have 

x + y=y  + x,  (3) 

an  equation  which  would  equally  hold  true  if  x and  y represented 
numbers , and  + were  the  sign  of  arithmetical  addition. 

Let  the  symbol  z stand  for  the  adjective  “European,”  then 
since  it  is,  in  effect,  the  same  thing  to  say  “ European  men  and 
women,”  as  to  say  “ European  men  and  European  women,”  we 
have 

z (x  + ij)  = zx  + zy.  (4) 

And  this  equation  also  would  be  equally  true  were  x,  y,  and  z 
symbols  of  number,  and  were  the  juxtaposition  of  two  literal 
symbols  to  represent  their  algebraic  product,  just  as  in  the  logical 
signification  previously  given,  it  represents  the  class  of  objects  to 
which  both  the  epithets  conjoined  belong. 

The  above  are  the  laws  which  govern  the  use  of  the  sign 
+,  here  used  to  denote  the  positive  operation  of  aggregating 
parts  into  a whole.  But  the  very  idea  of  an  operation  effecting 
some  positive  change  seems  to  suggest  to  us  the  idea  of  an  oppo- 
site or  negative  operation,  having  the  effect  of  undoing  what  the 
former  one  has  done.  Thus  wre  cannot  conceive  it -possible  to 
collect  parts  into  a whole,  and  not  conceive  it  also  possible  to 
separate  a part  from  a whole.  This  operation  we  express  in 
common  language  by  the  sign  except , as,  “ All  men  except 
Asiatics,”  “ All  states  except  those  which  are  monarchical.” 
Here  it  is  implied  that  the  things  excepted  form  a part  of  the 
things  from  which  they  are  excepted.  As  we  have  expressed 
the  operation  of  aggregation  by  the  sign  +,  so  we  may  express 
the  negative  operation  above  described  by  - minus.  Thus  if  x 
be  taken  to  represent  men,  and  y,  Asiatics,  i.  e.  Asiatic  men, 


34 


SIGNS  AND  THEIR  LAWS. 


[CHAP.  II. 

then  the  conception  of  “All  men  except  Asiatics”  will  be  ex- 
pressed by  x - y.  And  if  we  represent  by  x,  “ states,”  and  by 
y the  descriptive  property  “ having  a monarchical  form,”  then 
the  conception  of  “ All  states  except  those  which  are  monarchi- 
cal” will  be  expressed  by  x - xy. 

As  it  is  indifferent  for  all  the  essential  purposes  of  reasoning 
whether  we  express  excepted  cases  first  or  last  in  the  order  of 
speech,  it  is  also  indifferent  in  what  order  we  write  any  series  of 
terms,  some  of  which  are  affected  by  the  sign  -.  Thus  we  have, 
as  in  the  common  algebra, 

x - y = - y + x.  (5) 

Still  representing  by  x the  class  “men,”  and  by  y “Asiatics,” 
let  z represent  the  adjective  “ white.”  Now  to  apply  the  adjec- 
tive “white”  to  the  collection  of  men  expressed  by  the  phrase 
“ Men'  except  Asiatics,”  is  the  same  as  to  say,  “ White  men, 
except  white  Asiatics.”  Hence  we  have 

z(p  — y)  - zx  - zy . (6) 

This  is  also  in  accordance  with  the  laws  of  ordinary  algebra. 

The  equations  (4)  and  (6)  may  be  considered  as  exemplifica- 
tion of  a single  general  law,  which  may  be  stated  by  saying,  that 
the  literal  symbols,  x,  y,  z,  Sfc.  are  distributive  in  their  operation. 
The  general  fact  which  that  law  expresses  is  this,  viz. : — If  any 
quality  or  circumstance  is  ascribed  to  all  the  members  of  a group, 
formed  either  by  aggregation  or  exclusion  of  partial  groups,  the 
resulting  conception  is  the  same  as  if  the  quality  or  circumstance 
were  first  ascribed  to  each  member  of  the  partial  groups,  and  the 
aggregation  or  exclusion  effected  afterwards.  That  which  is 
ascribed  to  the  members  of  the  whole  is  ascribed  to  the  members 
of  all  its  parts,  howsoever  those  parts  are  connected  together. 

CLASS  III, 

12.  Signs  by  which  relation  is  expressed,  and  by  which  we 
form  propositions. 

Though  all  verbs  may  with  propriety  be  referred  to  this  class, 
it  is  sufficient  for  the  purposes  of  Logic  to  consider  it  as  includ- 
ing only  the  substantive  verb  is  or  are,  since  every  other  verb 


CHAP.  II.]  SIGNS  AND  THEIR  LAWS.  3 5 

may  be  resolved  into  this  element,  and  one  of  the  signs  included 
under  Class  i.  For  as  those  signs  are  used  to  express  quality  or 
circumstance  of  every  kind,  they  may  be  employed  to  express 
the  active  or  passive  relation  of  the  subject  of  the  verb,  considered 
with  reference  either  to  past,  to  present,  or  to  future  time. 
Thus  the  Proposition,  “ Caesar  conquered  the  Gauls,”  may  be 
resolved  into  “ Caesar  is  he  who  conquered  the  Gauls.”  The 
ground  of  this  analysis  I conceive  to  be  the  following : — Unless 
we  understand  what  is  meant  by  having  conquered  the  Gauls, 
i.  e.  by  the  expression  “ One  who  conquered  the  Gauls,”  we 
cannot  understand  the  sentence  in  question.  It  is,  therefore, 
truly  an  element  of  that  sentence ; another  element  is  “ Caesar,” 
and  there  is  yet  another  required,  the  copula  is,  to  show  the 
connexion  of  these  two.  I do  not,  however,  affirm  that  there  is 
no  other  mode  than  the  above  of  contemplating  the  relation  ex- 
pressed by  the  proposition,  “ Caesar  conquered  the  Gauls;”  but 
only  that  the  analysis  here  given  is  a correct  one  for  the  particu- 
lar point  of  view  which  has  been  taken,  and  that  it  suffices  for 
the  purposes  of  logical  deduction.  It  may  be  remarked  that  the 
passive  and  future  participles  of  the  Greek  language  imply  the 
existence  of  the  principle  which  has  been  asserted,  viz. : that  the 
sign  is  or  are  may  be  regarded  as  an  element  of  every  personal 
verb. 

13.  The  above  sign,  is  or  are,  may  be  expressed  by  the  sym- 
bol =.  The  laws,  or  as  would  usually  be  said,  the  axioms  which 
the  symbol  introduces,  are  next  to  be  considered. 

Let  us  take  the  Proposition,  “ The  stars  are  the  suns  and  the 
planets,”  and  let  us  represent  stars  by  x,  suns  by  y,  and  planets 
by  z;  we  have  then 

x = y + z.  (7) 

Now  if  it  be  true  that  the  stars  are  the  suns  and  the  planets,  it 
will  follow  that  the  stars,  except  the  planets,  are  suns.  This 
would  give  the  equation 

x-  z=y,  (8) 

which  must  therefore  be  a deduction  from  (7).  Thus  a term  z 
has  been  removed  from  one  side  of  an  equation  to  the  other  by 


36  SIGNS  AND  THEIR  LAWS.  [CHAP.  II. 

changing  its  sign.  This  is  in  accordance  with  the  algebraic  rule 
of  transposition. 

But  instead  of  dwelling  upon  particular  cases,  we  may  at  once 
affirm  the  general  axioms  : — 

1st.  If  equal  things  are  added  to  equal  things,  the  wholes  are 
equal. 

2nd.  If  equal  things  are  taken  from  equal  things,  the  re- 
mainders are  equal. 

And  it  hence  appears  that  we  may  add  or  subtract  equations, 
and  employ  the  rule  of  transposition  above  given  just  as  in  com- 
mon algebra. 

Again  : If  two  classes  of  things,  x and  y,  be  identical,  that  is, 
if  all  the  members  of  the  one  are  members  of  the  other,  then 
those  members  of  the  one  class  which  possess  a given  property  z 
will  be  identical  with  those  members  of  the  other  which  possess 
the  same  property  z.  Hence  if  we  have  the  equation 

x = y \ 

then  whatever  class  or  property  z may  represent,  we  have  also 

zx  - zy. 

This  is  formally  the  same  as  the  algebraic  law  : — If  both  mem- 
bers of  an  equation  are  multiplied  by  the  same  quantity,  the 
products  are  equal. 

In  like  manner  it  may  be  shown  that  if  the  corresponding 
members  of  two  equations  are  multiplied  together,  the  resulting 
equation  is  true. 

14.  Here,  however,  the  analogy  of  the  present  system  with 
that  of  algebra,  as  commonly  stated,  appears  to  stop.  Suppose  it 
true  that  those  members  of  a class  x which  possess  a certain  pro- 
perty z are  identical  with  those  members  of  a class  y which  pos- 
sess the  same  property  z,  it  does  not  follow  that  the  members  of 
the  class  x universally  are  identical  with  the  members  of  the 
class  y.  Hence  it  cannot  be  inferred  from  the  equation 

zx  = zy , 

that  the  equation 

x = y 

is  also  true.  In  other  words,  the  axiom  of  algebraists,  that  both 


CHA1>.  II.] 


SIGNS  AND  THEIR  LAWS. 


37 


sides  of  an  equation  may  be  divided  by  the  same  quantity,  has  no 
formal  equivalent  here.  I say  no  formal  equivalent,  because,  in 
accordance  with  the  general  spirit  of  these  inquiries,  it  is  not 
even  sought  to  determine  whether  the  mental  operation  which  is 
represented  by  removing  a logical  symbol,  z,  from  a combination 
zx,  is  in  itself  analogous  with  the  operation  of  division  in  Arith- 
metic. That  mental  operation  is  indeed  identical  with  what  is 
commonly  termed  Abstraction,  and  it  will  hereafter  appear  that 
its  laws  are  dependent  upon  the  laws  already  deduced  in  this 
chapter.  What  has  now  been  shown  is,  that  there  does  not 
exist  among  those  laws  anything  analogous  in  form  with  a com- 
monly received  axiom  of  Algebra. 

But  a little  consideration  will  show  that  even  in  common 
algebra  that  axiom  does  not  possess  the  generality  of  those  other 
axioms  Avhich  have  been  considered.  The  deduction  of  the 
equation  x = y from  the  equation  zx  = zy  is  only  valid  when  it 
is  known  that  z is  not  equal  to  0.  If’  then  the  value  z = 0 is 
supposed  to  be  admissible  in  the  algebraic  system,  the  axiom 
above  stated  ceases  to  be  applicable,  and  the  analogy  before  ex- 
emplified remains  at  least  unbroken. 

15.  However,  it  is  not  with  the  symbols  of  quantity  generally 
that  it  is  of  any  importance,  except  as  a matter  of  speculation,  to 
trace  such  affinities.  We  have  seen  (II.  9)  that  the  symbols  of 
Logic  are  subject  to  the  special  law, 

x-  = x. 

Now  of  the  symbols  of  Number  there  are  but  two,  viz.  0 and  1, 
which  are  subject  to  the  same  formal  lawr.  We  know  that  02  = 0, 
and  that  12  = 1 ; and  the  equation  :c2  = x,  considered  as  algebraic, 
has  no  other  roots  than  0 and  1 . Hence,  instead  of  determining 
the  measure  of  formal  agreement  of  the  symbols  of  Logic  with 
those  of  Number  generally,  it  is  more  immediately  suggested  to 
us  to  compare  them  with  symbols  of  quantity  admitting  only  of 
the  values  0 and  1.  Let  us  conceive,  then,  of  an  Algebra  in 
which  the  symbols  x,  y,  z,  &c.  admit  indifferently  of  the  values 
0 and  1,  and  of  these  values  alone.  The  laws,  the  axioms,  and 
the  processes,  of  such  an  Algebra  will  be  identical  in  their  whole 
extent  with  the  laws,  the  axioms,  and  the  processes  of  an  Al- 


38 


SIGNS  AND  THEIR  LAWS. 


[CHAP.  II. 


gebra  of  Logic.  Difference  of  interpretation  will  alone  divide 
them.  Upon  this  principle  the  method  of  the  following  work  is 
established. 

16.  It  now  remains  to  show  that  those  constituent  parts  of 
ordinary  language  which  have  not  been  considered  in  the  pre- 
vious sections  of  this  chapter  are  either  resolvable  into  the  same 
elements  as  those  which  have  been  considered,  or  are  subsidiary 
to  those  elements  by  contributing  to  their  more  precise  defi- 
nition. 

The  substantive,  the  adjective,  and  the  verb,  together  with 
the  particles  and , except,  we  have  already  considered.  The  pro- 
noun may  be  regarded  as  a particular  form  of  the  substantive  or 
the  adjective.  The  adverb  modifies  the  meaning  of  the  verb,  but 
does  not  affect  its  nature.  Prepositions  contribute  to  the  ex- 
pression of  circumstance  or  relation,  and  thus  tend  to  give  pre- 
cision and  detail  to  the  meaning  of  the  literal  symbols.  The 
conjunctions  if,  either,  or,  are  used  chiefly  in  the  expression  of 
relation  among  propositions,  and  it  will  hereafter  be  shown  that 
the  same  relations  can  be  completely  expressed  by  elementary 
symbols  analogous  in  interpretation,  and  identical  in  form  and 
law  with  the  symbols  whose  use  and  meaning  have  been  ex- 
plained in  this  Chapter.  As  to  any  remaining  elements  of 
speech,  it  will,  upon  examination,  be  found  that  they  are  used 
either  to  give  a more  definite  significance  to  the  terms  of  dis- 
course, and  thus  enter  into  the  interpretation  of  the  literal  sym- 
bols already  considered,  or  to  express  some  emotion  or  state  of 
feeling  accompanying  the  utterance  of  a proposition,  and  thus  do 
not  belong  to  the  province  of  the  understanding,  with  which 
alone  our  present  concern  lies.  Experience  of  its  use  will  tes- 
tify to  the  sufficiency  of  the  classification  which  has  been  adopted. 


CHAP.  III.] 


DERIVATION  OF  THE  LAWS. 


39 


CHAPTER  III. 

DERIVATION  OF  THE  LAWS  OF  THE  SYMBOLS  OF  LOGIC  FROM  THE 
LAWS  OF  THE  OPERATIONS  OF  THE  HUMAN  MIND. 

1.  r I ''HE  object  of  science,  properly  so  called,  is  the  knowledge 
oflaws  and  relations.  To  be  able  to  distinguish  what 
is  essential  to  this  end,  from  what  is  only  accidentally  associated 
with  it,  is  one  of  the  most  important  conditions  of  scientific  pro- 
gress. I say,  to  distinguish  between  these  elements,  because  a con- 
sistent devotion  to  science  does  not  require  that  the  attention 
should  be  altogether  withdrawn  from  other  speculations,  often  of  a 
metaphysical  nature,  with  which  it  is  not  unfrequently  connected. 
Such  questions,  for  instance,  as  the  existence  of  a sustaining 
ground  of  phenomena,  the  reality  of  cause,  the  propriety  of  forms 
of  speech  implying  that  the  successive  states  of  things  are  con- 
nected by  operations , and  others  of  a like  nature,  may  possess 
a deep  interest  and  significance  in  relation  to  science,  without 
being  essentially  scientific.  It  is  indeed  scarcely  possible  to 
express  the  conclusions  of  natural  science  without  borrowing 
the  language  of  these  conceptions.  Nor  is  there  necessarily 
any  practical  inconvenience  arising  from  this  source.  They  who 
believe,  and  they  who  refuse  to  believe,  that  there  is  more  in  the 
relation  of  cause  and  effect  than  an  invariable  order  of  succession, 
agree  in  their  interpretation  of  the  conclusions  of  physical  astro- 
nomy. But  they  only  agree  because  they  recognise  a common  ele- 
ment of  scientific  truth,  which  is  independent  of  their  particular 
views  of  the  nature  of  causation. 

2.  If  this  distinction  is  important  in  physical  science,  much 
more  does  it  deserve  attention  in  connexion  with  the  science  of 
the  intellectual  powers.  For  the  questions  which  this  science 
presents  become,  in  expression  at  least,  almost  necessarily  mixed 
up  with  modes  of  thought  and  language,  which  betray  a meta- 
physical origin.  The  idealist  would  give  to  the  laws  of  reasoning 


40  DERIVATION  OF  THE  LAWS.  [CHAP.  III. 

one  form  of  expression ; the  sceptic,  if  true  to  his  principles,  ano- 
ther. They  who  regard  the  phenomena  with  which  we  are  con- 
cerned in  this  inquiry  as  the  mere  successive  states  of  the  thinking 
subject  devoid  of  any  causal  connexion,  and  they  who  refer  them 
to  the  operations  of  an  active  intelligence,  would,  if  consistent, 
equally  differ  in  their  modes  of  statement.  Like  difference  would 
also  result  from  a difference  of  classification  of  the  mental  faculties. 
Now  the  principle  which  I would  here  assert,  as  affording  us  the 
only  ground  of  confidence  and  stability  amid  so  much  of  seeming 
and  of  real  diversity,  is  the  following,  viz.,  that  if  the  laws  in  ques- 
tion are  really  deduced  from  observation,  they  have  a real  existence 
as  laws  of  the  human  mind,  independently  of  any  metaphysical 
theory  which  may  seem  to  be  involved  in  the  mode  of  their  state- 
ment. They  contain  an  element  of  truth  which  no  ulterior  cri- 
ticism upon  the  nature,  or  even  upon  the  reality,  of  the  mind’s 
operations,  can  essentially  affect.  Let  it  even  be  granted  that 
the  mind  is  but  a succession  of  states  of  consciousness,  a series 
of  fleeting  impressions  uncaused  from  without  or  from  within, 
emerging  out  of  nothing,  and  returning  into  nothing  again, — 
the  last  refinement  of  the  sceptic  intellect, — still,  as  laws  of  suc- 
cession, or  at  least  of  a past  succession,  the  results  to  which  obser- 
vation had  led  would  remain  true.  They  would  require  to  be 
interpreted  into  a language  from  whose  vocabulary  all  such  terms 
as  cause  and  effect,  operation  and  subject,  substance  and  attri- 
bute, had  been  banished ; but  they  would  still  be  valid  as  scien- 
tific truths. 

Moreover,  as  any  statement  of  the  laws  of  thought,  founded 
upon  actual  observation,  must  thus  contain  scientific  elements 
which  are  independent  of  metaphysical  theories  of  the  nature  of 
the  mind,  the  practical  application  of  such  elements  to  the  con- 
struction of  a system  or  method  of  reasoning  must  also  be  inde- 
pendent of  metaphysical  distinctions.  For  it  is  upon  the  scien- 
tific elements  involved  in  the  statement  of  the  laws,  that  any 
practical  application  will  rest,  just  as  the  practical  conclusions  of 
physical  astronomy  are  independent  of  any  theory  of  the  cause 
of  gravitation,  but  rest  only  on  the  knowledge  of  its  phasno- 
menal  effects.  And,  therefore,  as  respects  both  the  determi- 


DERIVATION  OF  THE  LAWS. 


41 


CHAP.  III.] 

nation  of  the  laws  of  thought,  and  the  practical  use  of  them 
when  discovered,  we  are,  for  all  really  scientific  ends,  uncon- 
cerned with  the  truth  or  falsehood  of  any  metaphysical  specula- 
tions whatever. 

3.  The  course  which  it  appears  to  me  to  be  expedient,  under 
these  circumstances,  to  adopt,  is  to  avail  myself  as  far  as  possible 
of  the  language  of  common  discourse,  without  regard  to  any 
theory  of  the  nature  and  powers  of  the  mind  which  it  may  be 
thought  to  embody.  For  instance,  it  is  agreeable  to  common 
usage  to  say  that  we  converse  with  each  other  by  the  communi- 
cation of  ideas,  or  conceptions,  such  communication  being  the 
office  of  words ; and  that  with  reference  to  any  particular  ideas  or 
conceptions  presented  to  it,  the  mind  possesses  certain  powers  or 
faculties  by  which  the  mental  regard  maybe  fixed  upon  some  ideas, 
to  the  exclusion  of  others,  or  by  which  the  given  conceptions  or 
ideas  may,  in  various  ways,  be  combined  together.  To  those 
faculties  or  powers  different  names,  as  Attention,  Simple  Appre- 
hension, Conception  or  Imagination,  Abstraction,  &c.,  have  been 
given, — names  which  have  not  only  furnished  the  titles  of  distinct 
divisions  of  the  philosophy  of  the  human  mind,  but  passed  into 
the  common  language  of  men.  Whenever,  then,  occasion  shall 
occur  to  use  these  terms,  I shall  do  so  without  implying  thereby 
that  I accept  the  theory  that  the  mind  possesses  such  and  such 
powers  and  faculties  as  distinct  elements  of  its  activity.  Nor  is 
it  indeed  necessary  to  inquire  whether  such  powers  of  the  under- 
standing have  a distinct  existence  or  not.  We  may  merge  these 
different  titles  under  the  one  generic  name  of  Operations  of  the 
human  mind,  define  these  operations  so  far  as  is  necessary  for  the 
purposes  of  this  work,  and  then  seek  to  express  their  ultimate  laws. 
Such  will  be  the  general  order  of  the  course  which  I shall  pur- 
sue, though  reference  will  occasionally  be  made  to  the  names  which 
common  agreement  has  assigned  to  the  particular  states  or  ope- 
rations of  the  mind  which  may  fall  under  our  notice. 

It  will  be  most  convenient  to  distribute  the  more  definite  re- 
sults of  the  following  investigation  into  distinct  Propositions. 


42 


DERIVATION  OF  THE  LAWS. 


[CHAP.  III. 


Proposition  I. 

4.  To  deduce  the  laws  of  the  symbols  of  Logic  from  a conside- 
ration of  those  operations  of  the  mind  which  are  implied  in  the  strict 
use  of  language  as  an  instrument  of  reasoning. 

In  every  discourse,  whether  of  the  mind  conversing  with  its 
own  thoughts,  or  of  the  individual  in  his  intercourse  with  others, 
there  is  an  assumed  or  expressed  limit  within  which  the  subjects  of 
its  operation  are  confined.  The  most  unfettered  discourse  is  that 
in  which  the  words  we  use  are  understood  in  the  widest  possible 
application,  and  for  them  the  limits  of  discourse  are  co-extensive 
with  those  of  the  universe  itself.  But  more  usually  we  confine  our- 
selves to  a less  spacious  field.  Sometimes,  in  discoursing  of  men 
we  imply  (without  expressing  the  limitation)  that  it  is  of  men 
only  under  certain  circumstances  and  conditions  that  we  speak, 
as  of  civilized  men,  or  of  men  in  the  vigour  of  life,  or  of  men 
under  some  other  condition  or  relation.  Now,  whatever  maybe 
the  extent  of  the  field  within  which  all  the  objects  of  our  dis- 
course are  found,  that  field  may  properly  be  termed  the  universe 
of  discourse. 

5.  Furthermore,  this  universe  of  discourse  is  in  the  strictest 
sense  the  ultimate  subject  of  the  discourse.  The  office  of  any  name 
or  descriptive  term  employed  under  the  limitations  supposed  is  not 
to  raise  in  the  mind  the  conception  of  all  the  beings  or  o bjects  to 
which  that  name  or  description  is  applicable,  but  only  of  those 
which  exist  within  the  supposed  universe  of  discourse.  If  that 
universe  of  discourse  is  the  actual  universe  of  things,  which  it 
always  is  when  our  words  are  taken  in  their  real  and  literal  sense, 
then  by  men  we  mean  all  men  that  exist ; but  if  the  universe  of 
discourse  is  limited  by  any  antecedent  implied  understanding, 
then  it  is  of  men  under  the  limitation  thus  introduced  that  we 
speak.  It  is  in  both  cases  the  business  of  the  word  men  to  direct 
a certain  operation  of  the  mind,  by  which,  from  the  proper  uni- 
verse of  discourse,  we  select  or  fix  upon  the  individuals  signified. 

6.  Exactly  of  the  same  kind  is  the  mental  operation  implied 
by  the  use  of  an  adjective.  Let,  for  instance,  the  universe  of  dis- 
course be  the  actual  Universe.  Then,  as  the  word  men  directs 


CHAP.  III.]  DERIVATION  OF  THE  LAWS.  43 

us  to  select  mentally  from  that  Universe  all  the  beings  to  which 
the  term  “men”  is  applicable  ; so  the  adjective  “good,”  in  the 
combination  “ good  men,”  directs  us  still  further  to  select  men- 
tally from  the  class  of  men  all  those  who  possess  the  further 
quality  “good;”  andfff  another  adjective  were  prefixed  to  the 
combination  “ good  men,”  it  would  direct  a further  operation  of 
the  same  nature,  having  reference  to  that  further  quality  which 
it  might  be  chosen  to  express. 

It  is  important  to  notice  carefully  the  real  nature  of  the  ope- 
ration here  described,  for  it  is  conceivable,  that  it  might  have 
been  different  from  what  it  is.  Were  the  adjective  simply  attri- 
butive in  its  character,  it  would  seem,  that  when  a particular  set 
of  beings  is  designated  by  men , the  prefixing  of  the  adjective 
good  would  direct  us  to  attach  mentally  to  all  those  beings  the 
quality  of  goodness.  But  this  is  not  the  real  office  of  the  ad- 
jective. The  operation  which  we  really  perform  is  one  of  se- 
lection according  to  a prescribed  principle  or  idea.  To  what  fa- 
culties of  the  mind  such  an  operation  would  be  referred,  according 
to  the  received  classification  of  its  powers,  it  is  not  important  to 
inquire,  but  I suppose  that  it  would  be  considered  as  dependent 
upon  the  two  faculties  of  Conception  or  Imagination,  and  Atten- 
tion. To  the  one  of  these  faculties  might  be  referred  the  forma- 
tion of  the  general  conception  ; to  the  other  the  fixing  of  the 
mental  regard  upon  those  individuals  within  the  prescribed  uni- 
verse of  discourse  which  answer  to  the  conception.  If,  however, 
as  seems  not  improbable,  the  power  of  Attention  is  nothing  more 
than  the  power  of  continuing  the  exercise  of  any  other  faculty  of  the 
mind,  we  might  properly  regard  the  whole  of  the  mental  process 
above  described  as  referable  to  the  mental  faculty  of  Imagination 
or  Conception,  the  first  step  of  the  process  being  the  conception 
of  the  Universe  itself,  and  each  succeeding  step  limiting  in  a de- 
finite manner  the  conception  thus  formed.  Adopting  this  view,  I 
shall  describe  each  such  step,  or  any  definite  combination  of  such 
steps,  as  a definite  act  of  conception.  And  the  use  of  this  term  I 
shall  extend  so  as  to  include  in  its  meaning  not  only  the  conception 
of  classes  of  objects  represented  by  particular  names  or  simple 
attributes  of  quality,  but  also  the  combination  of  such  concep- 
tions in  any  manner  consistent  with  the  powers  and  limitations 


44 


DERIVATION  OF  THE  LAWS. 


[CHAP.  III. 

of  the  human  mind ; indeed,  any  intellectual  operation  short 
of  that  which  is  involved  in  the  structure  of  a sentence  or  propo- 
sition. The  general  laws  to  which  such  operations  of  the  mind 
are  subject  are  now  to  be  considered. 

7.  Now  it  will  be  shown  that  the  laws  which  in  the  preced- 
ing chapter  have  been  determined  a posteriori  from  the  consti- 
tution of  language,  for  the  use  of  the  literal  symbols  of  Logic, 
are  in  reality  the  laws  of  that  definite  mental  operation  which 
has  just  been  described.  We  commence  our  discourse  with  a 
certain  understanding  as  to  the  limits  of  its  subject,  i.  e.  as  to 
the  limits  of  its  Universe.  Every  name,  every  term  of  descrip- 
tion that  we  employ,  directs  him  Avhom  we  address  to  the  per- 
formance of  a certain  mental  operation  upon  that  subject.  And 
thus  is  thought  communicated.  But  as  each  name  or  descriptive 
term  is  in  this  view  but  the  representative  of  an  intellectual  ope- 
ration, that  operation  being  also  prior  in  the  order  of  nature,  it 
is  clear  that  the  laws  of  the  name  or  symbol  must  be  of  a deriva- 
tive character,— must,  in  fact,  originate  in  those  of  the  operation 
which  they  represent.  That  the  laws  of  the  symbol  and  of  the 
mental  process  are  identical  in  expression  will  now  be  shown. 

8.  Let  us  then  suppose  that  the  universe  of  our  discourse  is 
the  actual  universe,  so  that  words  are  to  be  used  in  the  full  ex- 
tent of  their  meaning,  and  let  us  consider  the  two  mental  opera- 
tions implied  by  the  words  “white”  and  “men.”  The  word 
“ men”  implies  the  operation  of  selecting  in  thought  from  its 
subject,  the  universe,  all  men;  and  the  resulting  conception, 
men , becomes  the  subject  of  the  next  operation.  The  operation 
implied  by  the  word  “ white”  is  that  of  selecting  from  its  subject, 
“ men,”  all  of  that  class  which  are  white.  The  final  resulting 
conception  is  that  of  “white  men.”  Now  it  is  perfectly  appa- 
rent that  if  the  operations  above  described  had  been  performed 
in  a converse  order,  the  result  would  have  been  the  same.  Whe- 
ther we  begin  by  forming  the  conception  of  “»zen,”  and  then 
by  a second  intellectual  act  limit  that  conception  to  “ white 
men,”  or  whether  we  begin  by  forming  the  conception  of  “ white 
objects,”  and  then  limit  it  to  such  of  that  class  as  are  “men,”  is 
perfectly  indifferent  so  far  as  the  result  is  concerned.  It  is  ob- 
vious that  the  order  of  the  mental  processes  would  be  equally 


CHAP.  III.] 


DERIVATION  OF  THE  LAWS. 


45 


indifferent  if  for  the  words  “white”  and  “men”  we  substituted 
any  other  descriptive  or  appellative  terms  whatever,  provided 
only  that  their  meaning  was  fixed  and  absolute.  And  thus  the 
indifference  of  the  order  of  two  successive  acts  of  the  faculty  of 
Conception,  the  one  of  which  furnishes  the  subject  upon  which 
the  other  is  supposed  to  operate,  is  a general  condition  of  the 
exercise  of  that  faculty.  It  is  a law  of  the  mind,  and  it  is  the 
real  origin  of  that  law  of  the  literal  symbols  of  Logic  which  con- 
stitutes its  formal  expression  (1)  Chap.  n. 

9.  It  is  equally  clear  that  the  mental  operation  above  de- 
scribed is  of  such  a nature  that  its  effect  is  not  altered  by  repe- 
tition. Suppose  that  by  a definite  act  of  conception  the  attention 
has  been  fixed  upon  men,  and  that  by  another  exercise  of  the 
sam'e  faculty  we  limit  it  to  those  of  the  race  who  are  white. 
Then  any  further  repetition  of  the  latter  mental  act,  by  which 
the  attention  is  limited  to  white  objects,  does  not  in  any  way 
modify  the  conception  arrived  at,  viz.,  that  of  ’white  men.  This 
is  also  an  example  of  a general  law  of  the  mind,  and  it  has  its 
formal  expression  in  the  law  ((2)  Chap,  n.)  of  the  literal  symbols. 

10.  Again,  it  is  manifest  that  from  the  conceptions  of  two 
distinct  classes  of  things  we  can  form  the  conception  of  that  col- 
lection of  things  which  the  two  classes  taken  together  compose ; 
and  it  is  obviously  indifferent  in  what  order  of  position  or  of 
priority  those  classes  are  presented  to  the  mental  view.  This  is 
another  general  law  of  the  mind,  and  its  expression  is  found  in 
(3)  Chap.  ii. 

11.  It  is  not  necessary  to  pursue  this  course  of  inquiry  and 
comparison.  Sufficient  illustration  has  been  given  to  render  ma- 
nifest the  two  following  positions,  viz.  : 

First,  That  the  operations  of  the  mind,  by  -which,  in  the 
exercise  of  its  power  of  imagination  or  conception,  it  combines 
and  modifies  the  simple  ideas  of  things  or  qualities,  not  less  than 
those  operations  of  the  reason  which  are  exercised  upon  truths 
and  propositions,  are  subject  to  general  laws. 

Secondly,  That  those  laws  are  mathematical  in  their  form, 
and  that  they  are  actually  developed  in  the  essential  lawrs  of 
human  language;  Wherefore  the  laws  of  the  symbols  of  Logic 


46  DERIVATION  OF  THE  LAW'S.  [CHAP.  III. 

are  deducible  from  a consideration  of  the  operations  of  the  mind 
in  reasoning. 

12.  The  remainder  of  this  chapter  will  be  occupied  with 
questions  relating  to  that  law  of  thought  whose  expression  is 
x2  = x (II.  9),  a law  which,  as  has  been  implied  (II.  15),  forms 
the  characteristic  distinction  of  the  operations  of  the  mind  in  its 
ordinary  discourse  and  reasoning,  as  compared  with  its  operations 
when  occupied  with  the  general  algebra  of  quantity.  An  im- 
portant part  of  the  following  inquiry  will  consist  in  proving  that 
the  symbols  0 and  1 occupy  a place,  and  are  susceptible  of  an 
interpretation,  among  the  symbols  of  Logic  ; and  it  may  first  be 
necessary  to  show  how  particular  symbols,  such  as  the  above, 
may  with  propriety  and  advantage  be  employed  in  the  represen- 
tation of  distinct  systems  of  thought. 

The  ground  of  this  propriety  cannot  consist  in  any  commu- 
nity of  interpretation.  For  in  systems  of  thought  so  truly 
distinct  as  those  of  Logic  and  Arithmetic  (I  use  the  latter  term 
in  its  widest  sense  as  the  science  of  Number),  there  is,  properly 
speaking,  no  community  of  subject.  The  one  of  them  is  conver- 
sant with  the  very  conceptions  of  things,  the  other  takes  account 
solely  of  their  numerical  relations.  But  inasmuch  as  the  forms 
and  methods  of  any  system  of  reasoning  depend  immediately  upon 
the  laws  to  which  the  symbols  are  subject,  and  only  mediately, 
through  the  above  link  of  connexion,  upon  their  interpretation, 
there  may  be  both  propriety  and  advantage  in  employing  the 
same  symbols  in  different  systems  of  thought,  provided  that  such 
interpretations  can  be  assigned  to  them  as  shall  render  their  for- 
mal laws  identical,  and  their  use  consistent.  The  ground  of  that 
employment  will  not  then  be  community  of  interpretation,  but 
the  community  of  the  formal  laws,  to  which  in  their  respective 
systems  they  are  subject.  Nor  must  that  community  of  formal 
laws  be  established  upon  any  other  ground  than  that  of  a careful 
observation  and  comparison  of  those  results  which  are  seen  to 
flow  independently  from  the  interpretations  of  the  systems  under 
consideration. 

These  observations  will  explain  the  process  of  inquiry  adopted 
in  the  following  Proposition.  The  literal  symbols  of  Logic  are 


CHAP.  III.]  DERIVATION  OF  tHE  LAWS.  47 

universally  subject  to  the  law  whose  expression  is  x 2 = x.  Of 
the  symbols  of  Number  there  are  two  only,  0 and  1,  which  sa- 
tisfy this  law.  But  each  of  these  symbols  is  also  subject  to  a law 
peculiar  to  itself  in  the  system  of  numerical  magnitude,  and  this 
suggests  the  inquiry,  what  interpretations  must  be  given  to  the 
literal  symbols  of  Logic,  in  order  that  the  same  peculiar  and 
formal  laws  may  be  realized  in  the  logical  system  also. 

Proposition  II. 

13.  To  determine  the  logical  value  and  significance  of  the 
symbols  0 and  1. 

The  symbol  0,  as  used  in  Algebra,  satisfies  the  following  for- 
mal law, 

0 x y = 0,  or  Qy  = 0,  (1) 

whatever  number  y may  represent.  That  this  formal  law  may  be 
obeyed  in  the  system  of  Logic,  we  must  assign  to  the  symbol  0 
such  an  interpretation  that  the  class  represented  by  0 y may  be 
identical  with  the  class  represented  by  0,  whatever  the  class  y 
may  be.  A little  consideration  will  show  that  this  condition  is 
satisfied  if  the  symbol  0 represent  Nothing.  In  accordance  with 
a previous  definition,  we  may  term  Nothing  a class.  In  fact, 
Nothing  and  Universe  are  the  two  limits  of  class  extension,  for 
they  are  the  limits  of  the  possible  interpretations  of  general 
names,  none  of  which  can  relate  to  fewer  individuals  than  are 
comprised  in  Nothing,  or  to  more  than  are  comprised  in  the 
Universe.  Now  whatever  the  class  y may  be,  the  individuals 
which  are  common  to  it  and  to  the  class  “ Nothing”  are  identi- 
cal with  those  comprised  in  the  class  “ Nothing,”  for  they  are 
none.  And  thus  by  assigning  to  0 the  interpretation  Nothing, 
the  law  ( 1 ) is  satisfied ; and  it  is  not  otherwise  satisfied  consis- 
tently with  the  perfectly  general  character  of  the  class  y. 

Secondly,  The  symbol  1 satisfies  in  the  system  of  Number 
the  following  law,  viz., 

1 x y = y>  or  1 y = y, 

whatever  number  y may  represent.  And  this  formal  equation 
being  assumed  as  equally  valid  in  the  system  of  this  work,  in 


48  DERIVATION  OF  THE  LAWS.  [CHAP.  III. 

which  1 and  y represent  classes,  it  appears  that  the  symbol  1 
must  represent  such  a class  that  all  the  individuals  which  are 
found  in  any  proposed  class  y are  also  all  the  individuals  1 y that 
are  common  to  that  class  y and  the  class  represented  by  1.  A' 
little  consideration  will  here  show  that  the  class  represented  by  1 
must  be  “ the  Universe,”  since  this  is  the  only  class  in  which 
are  found  all  the  individuals  that  exist  in  any  class.  Hence  the 
respective  interpretations  of  the  symbols  0 and  1 in  the  system 
of  Logic  are  Nothing  and  Universe. 

14.  As  with  the  idea  of  any  class  of  objects  as  “ men,”  there 
is  suggested  to  the  mind  the  idea  of  the  contrary  class  of  beings 
Avhich  are  not  men ; and  as  the  whole  Universe  is  made  up  of 
these  two  classes  together,  since  of  every  individual  which  it 
comprehends  we  may  affirm  either  that  it  is  a man,  or  that  it  is 
not  a man,  it  becomes  important  to  inquire  how  such  contrary 
names  are  to  be  expressed.  Such  is  the  object  of  the  following 
Proposition. 

Proposition  III. 

If  x represent  any  class  of  objects,  then  will  1 - x represent  the 
contrary  or  supplementary  class  of  objects.,  i.  e.  the  class  including 
all  objects  which  are  not  comprehended  in  the  class  x. 

F or  greater  distinctness  of  conception  let  x represent  the  class 
men , and  let  us  express,  according  to  the  last  Proposition,  the 
Universe  by  1 ; now  if  from  the  conception  of  the  Universe,  as 
consisting  of  “ men”  and  “not-men,”  we  exclude  the  conception 
of  “ men,”  the  resulting  conception  is  that  of  the  contrary  class, 

“ not-men.”  Hence  the  class  “ not-men”  will  be  represented  by 
1 - x.  And,  in  general,  whatever  class  of  objects  is  represented 
by  the  symbol  x,  the  contrary  class  will  be  expressed  by  1 - x. 

15.  Although  the  following  Proposition  belongs  in  strictness 
to  a future  chapter  of  this  work,  devoted  to  the  subject  of 
maxims  or  necessary  truths,  yet,  on  account  of  the  great  impor- 
tance of  that  law  of  thought  to  which  it  relates,  it  has  been 
thought  proper  to  introduce  it  here. 


CHAP.  III.] 


DERIVATION  OF  THE  LAWS. 


49 


Proposition  IV. 

That  axiom  of  metaphysicians  which  is  termed  the  principle  of 
contradiction , and  which  affirms  that  it  is  impossible  for  any  being  to 
possess  a quality,  and  at  the  same  time  not  to  possess  it,  is  a conse- 
quence of  the  fundamental  law  of  thought , whose  expression  is  x2  = x. 

Let  us  write  this  equation  in  the  form 
x - x2  = 0, 

whence  we  have 

x(\-x)  = 0;  (1) 

both  these  transformations  being  justified  by  the  axiomatic  laws 
of  combination  and  transposition  (II.  13).  Let  us,  for  simplicity 
of  conception,  give  to  the  symbol  x the  particular  interpretation 
of  men,  then  1 - x will  represent  the  class  of  “ not-men” 
(Prop,  iii.)  Now  the  formal  product  of  the  expressions  of  two 
classes  represents  that  class  of  individuals  which  is  common  to 
them  both  (II.  6).  Hence  #(1  - x)  will  represent  the  class 
whose  members  are  at  once  “men,”  and  “ not  men,”  and  the 
equation  (1)  thus  express  the  principle,  that  a class  whose  mem- 
bers are  at  the  same  time  men  and  not  men  does  not  exist.  In 
other  words,  that  it  is  impossible  for  the  same  individual  to  be  at 
the  same  time  a man  and  not  a man.  Now  let  the  meaning  of 
the  symbol  x be  extended  from  the  representing  of  “ men,”  to 
that  of  any  class  of  beings  characterized  by  the  possession  of  any 
quality  whatever;  and  the  equation  (1)  will  then  express  that  it 
is  impossible  for  a being  to  possess  a quality  and  not  to  possess 
that  quality  at  the  same  time.  But  this  is  identically  that 
“ principle  of  contradiction”  which  Aristotle  has  described  as  the 
fundamental  axiom  of  all  philosophy.  “ It  is  impossible  that  the 
same  quality  should  both  belong  and  not  belong  to  the  same 
thing.  . . This  is  the  most  certain  of  all  principles.  . . Wherefore 
they  who  demonstrate  refer  to  this  as  an  ultimate  opinion.  For 
it  is  by  nature  the  source  of  all  the  other  axioms.”* 


* To  yap  avro  apa  virapxtiv  rt  Kal  pi)  virapxtiv  aSvvarov  Tip  airip  Kai  Kara 
to  auTO.  . . A vty]  Si ) iraoGiv  lari  fitfiaiOTaTi]  tOiv  apxoiv.  . . A 16  Travrig  oi  cnroStiK- 
viivrtg  tig  TavTpv  avayovoiv  iaxaTi)v  S6£av  ipvcni  yap  apxn  Kai  tuiv  aWuw 
aZiuipaTwv  avTp  iravToiv — Metaphysica,  III.  3. 


50  DERIVATION  OF  THE  LAWS.  [CHAP.  III. 

The  above  interpretation  has  been  introduced  not  on  account 
of  its  immediate  value  in  the  present  system,  but  as  an  illustration 
of  a significant  fact  in  the  philosophy  of  the  intellectual  powers, 
viz.,  that  what  has  been  commonly  regarded  as  the  fundamental 
axiom  of  metaphysics  is  but  the  consequence  of  a law  of  thought, 
mathematical  in  its  form.  I desire  to  direct  attention  also  to  the 
circumstance  that  the  equation  (1)  in  which  that  fundamental 
law  of  thought  is  expressed  is  an  equation  of  the  second  degree.* * 
Without  speculating  at  all  in  this  chapter  upon  the  question, 
whether  that  circumstance  is  necessary  in  its  own  nature,  we 
may  venture  to  assert  that  if  it  had  not  existed,  the  whole  pro- 
cedure of  the  understanding  would  have  been  different  from  what 
it  is.  Thus  it  is  a consequence  of  the  fact  that  the  fundamental 
equation  of  thought  is  of  the  second  degree,  that  we  perform  the 
operation  of  analysis  and  classification,  by  division  into  pairs  of 


* Should  it  here  be  said  that  the  existence  of  the  equation  x2  = x necessitates 
also  the  existence  of  the  equation  x3  = x,  which  is  of  the  third  degree,  and  then 
inquired  whether  that  equation  does  not  indicate  a process  of  trichotomy ; the 
answer  is,  that  the  equation  x3  = x is  not  interpretable  in  the  system  of  logic. 
For  writing  it  in  either  of  the  forms 

* (1  - *)  (I  + *)  = 0,  (2) 

x (1  — x)  (—  1 — x)  = C,  (3) 

we  see  that  its  interpretation,  if  possible  at  all,  must  involve  that  of  the  factor 
1 + x,  or  of  the  factor  — 1 — x.  The  former  is  not  interpretable,  because  we 
cannot  conceive  of  the  addition  of  any  class  x to  the  universe  1 ; the  latter  is  not 

interpretable,  because  the  symbol  — 1 is  not  subject  to  the  law  x (1  — x)  = 0,  to 

which  all  class  symbols  are  subject.  Hence  the  equation  x3  = x admits  of  no  in- 
terpretation analogous  to  that  of  the  equation  x 2 = x.  Were  the  former  equation, 
however,  true  independently  of  the  latter,  i.  e.  were  that  act  of  the  mind  which 
is  denoted  by  the  symbol  x,  such  that  its  second  repetition  should  reproduce  the 
result  of  a single  operation,  but  not  its  first  or  merd  repetition,  it  is  presumable 
that  we  should  be  able  to  interpret  one  of  the  forms  (2),  (3),  which  under  the 
actual  conditions  of  thought  we  cannot  do.  There  exist  operations,  known  to 
the  mathematician,  the  law  of  which  may  be  adequately  expressed  by  the  equa- 
tion x3—x.  But  they  are  of  a nature  altogether  foreign  to  the  province  of 
general  reasoning. 

In  saying  that  it  is  conceivable  that  the  law  of  thought  might  have  been  dif- 
ferent from  what  it  is,  I mean  only  that  we  can  frame  such  an  hypothesis,  and 
study  its  consequences.  The  possibility  of  doing  this  involves  no  such  doctrine 
as  that  the  actual  law  of  human  reason  is  the  product  either  of  chance  or  of  arbi- 
trary will. 


CHAP.  III.]  DERIVATION  OF  THE  LAWS.  51 

opposites,  or,  as  it  is  technically  said,  by  dichotomy.  Now  if  the 
equation  in  question  had  been  of  the  third  degree,  still  admitting 
of  interpretation  as  such,  the  mental  division  must  have  been 
threefold  in  character,  and  we  must  have  proceeded  by  a species 
of  trichotomy,  the  real  nature  of  which  it  is  impossible  for  us, 
with  our  existing  faculties,  adequately  to  conceive,  but  the  laws 
of  which  we  might  still  investigate  as  an  object  of  intellectual 
speculation. 

16.  The  law  of  thought  expressed  by  the  equation  (1)  will, 
for  reasons  which  are  made  apparent  by  the  above  discussion,  be 
occasionally  referred  to  as  the  “ law  of  duality.” 


52 


DIVISION  OF  PROPOSITIONS. 


[CHAP.  IV. 


CHAPTER  IV. 

OF  THE  DIVISION  OF  PROPOSITIONS  INTO  THE  TWO  CLASSES  OF 
“PRIMARY”  AND  “ SECONDARY;”  OF  THE  CHARACTERISTIC  PRO- 
PERTIES OF  THOSE  CLASSES,  AND  OF  THE  LAWS  OF  THE  EXPRES- 
SION OF  PRIMARY  PROPOSITIONS. 

1 . r I ''HE  laws  of  t]iose  mental  operations  which  are  concerned 
in  the  processes  of  Conception  or  Imagination  having 
been  investigated,  and  the  corresponding  laws  of  the  symbols 
by  which  they  are  represented  explained,  we  are  led  to  consider 
the  practical  application  of  the  results  obtained : first,  in  the 
expression  of  the  complex  terms  of  propositions ; secondly,  in 
the  expression  of  propositions ; and  lastly,  in  the  construction  of 
a general  method  of  deductive  analysis.  In  the  present  chapter 
we  shall  be  chiefly  concerned  with  the  first  of  these  objects,  as 
an  introduction  to  which  it  is  necessary  to  establish  the  following 
Proposition : 

Proposition  I. 

All  logical  propositions  may  be  considered  as  belonging  to  one 
or  the  other  oftiuo  great  classes,  to  which  the  respective  names  of 
“ Primary"  or  “ Concrete  Propositions,"  and  “ Secondary ” or  “ Ab- 
stract Propositions,"  may  be  given. 

Every  assertion  that  we  make  may  be  referred  to  one  or  the 
other  of  the  two  following  kinds.  Either  it  expresses  a relation 
among  things,  or  it  expresses,  or  is  equivalent  to  the  expression  of, 
a relation  among  propositions.  An  assertion  respecting  the  pro- 
perties of  things,  or  the  phaenomena  which  they  manifest,  or  the 
circumstances  in  which  they  are  placed,  is,  properly  speaking,  the 
assertion  of  a relation  among  things.  To  say  that  “ snow  is 
white,”  is  for  the  ends  of  logic  equivalent  to  saying,  that  “snow 
is  a white  thing.”  An  assertion  respecting  facts  or  events,  their 
mutual  connexion  and  dependence,  is,  for  the  same  ends,  generally 
equivalent  to  the  assertion,  that  such  and  such  propositions  con- 


DIVISION  OF  PROPOSITIONS. 


53 


CHAP.  IV.] 

cerning  those  events  have  a certain  relation  to  each  other  as 
respects  their  mutual  truth  or  falsehood.  The  former  class  of 
propositions,  relating  to  things , I call  “ Primary the  latter  class, 
relating  to  propositions,  I call  “ Secondary.”  The  distinction  is 
in  practice  nearly  but  not  quite  co-extensive  with  the  common 
logical  distinction  of  propositions  as  categorical  or  hypothetical. 

For  instance,  the  propositions,  “The  sun  shines,”  “The  earth 
is  warmed,”  are  primary;  the  proposition,  “ If  the  sun  shines 
the  earth  is  warmed,”  is  secondary.  To  say,  “ The  sun  shines,” 
is  to  say,  “ The  sun  is  that  which  shines,”  and  it  expresses  a re- 
lation between  two  classes  of  things,  viz.,  “ the  sun”  and  “ things 
which  shine.”  The  secondary  proposition,  however,  given  above, 
expresses  a relation  of  dependence  between  the  two  primary  propo- 
sitions, “ The  sun  shines,”  and  “ The  earth  is  warmed.”  I do  not 
hereby  affirm  that  the  relation  between  these  propositions  is,  like 
that  which  exists  between  the  facts  which  they  express,  a rela- 
tion of  causality,  but  only  that  the  relation  among  the  propo- 
sitions so  implies,  and  is  so  implied  by,  the  relation  among  the 
facts,  that  it  may  for  the  ends  of  logic  be  used  as  a fit  repre- 
sentative of  that  relation. 

2.  If  instead  of  the  proposition,  “ The  sun  shines,”  we  say, 
“It  is  true  that  the  sun  shines,”  we  then  speak  not  directly  of 
things,  but  of  a proposition  concerning  things,  viz.,  of  the  pro- 
position, “ The  sun  shines.”  And,  therefore,  the  proposition  in 
which  we  thus  speak  is  a secondary  one.  Every  primary  pro- 
position may  thus  give  rise  to  a secondary  proposition,  viz.,  to 
that  secondary  proposition  which  asserts  its  truth,  or  declares  its 
falsehood. 

It  will  usually  happen,  that  the  particles  if,  either,  or,  will 
indicate  that  a proposition  is  secondary ; but  they  do  not  neces- 
sarily imply  that  such  is  the  case.  The  proposition,  “ Animals 
are  either  rational  or  irrational,”  is  primary.  It  cannot  be  re- 
solved into  “ Either  animals  are  rational  or  animals  are  irra- 
tional,” and  it  does  not  therefore  express  a relation  of  dependence 
between  the  two  propositions  connected  together  in  the  latter 
disjunctive  sentence.  The  particles,  either,  or,  are  in  fact  no 
criterion  of  the  nature  of  propositions,  although  it  happens  that 
they  are  more  frequently  found  in  secondary  propositions.  Even 


54  DIVISION  OF  PROPOSITIONS.  [CHAP.  IV. 

the  conjunction  if  may  be  found  in  primary  propositions.  “ Men 
are,  if  wise,  then  temperate,”  is  an  example  of  the  kind.  It 
cannot  be  resolved  into  “ If  all  men  are  wise,  then  all  men  are 
temperate.” 

3.  As  it  is  not  my  design  to  discuss  the  merits  or  defects  of 
the  ordinary  division  of  propositions,  I shall  simply  remark  here, 
that  the  principle  upon  which  the  present  classification  is  founded 
is  clear  and  definite  in  its  application,  that  it  involves  a real 
and  fundamental  distinction  in  propositions,  and  that  it  is  of 
essential  importance  to  the  development  of  a general  method  of 
reasoning.  Nor  does  the  fact  that  a primary  proposition  may 
be  put  into  a form  in  which  it  becomes  secondary  at  all  conflict 
with  the  views  here  maintained.  For  in  the  case  thus  supposed, 
it  is  not  of  the  things  connected  together  in  the  primary  propo- 
sition that  any  direct  account  is  taken,  but  only  of  the  propo- 
sition itself  considered  as  true  or  as  false. 

4.  In  the  expression  both  of  primary  and  of  secondary  propo- 
sitions, the  same  symbols,  subject,  as  it  will  appear,  to  the  same 
laws,  will  be  employed  in  this  work.  The  difference  between 
the  two  cases  is  a difference  not  of  form  but  of  interpretation. 
In  both  cases  the  actual  relation  which  it  is  the  object  of  the 
proposition  to  express  will  be  denoted  by  the  sign  =.  In  the 
expression  of  primary  propositions,  the  members  thus  connected 
will  usually  represent  the  “ terms”  of  a proposition,  or,  as  they 
are  more  particularly  designated,  its  subject  and  predicate. 

Proposition  II. 

5.  To  deduce  a general  method , founded  upon  the  enumeration  of 
possible  varieties,  for  the  expression  of  any  class  or  collection  of  things, 
which  may  constitute  a “ term'  of  a Primary  Proposition. 

First,  If  the  class  or  collection  of  things  to  be  expressed  is 
defined  only  by  names  or  qualities  common  to  all  the  individuals 
of  which  it  consists,  its  expression  will  consist  of  a single  term, 
in  which  the  symbols  expressive  of  those  names  or  qualities  will 
be  combined  without  any  connecting  sign,  as  if  by  the  alge- 
braic process  of  multiplication.  Thus,  if  x represent  opaque 
substances,  y polished  substances,  z stones,  we  shall  have, 


CHAP.  IV.]  DIVISION  OF  PROPOSITIONS.  55 

xyz  = opaque  polished  stones  ; 

xy  (1  - z)  = opaque  polished  substances  which  are  not  stones; 
x (1  - y)  (1  - z)  = opaque  substances  which  are  not  polished, 
and  are  not  stones  ; 

and  so  on  for  any  other  combination.  Let  it  be  observed,  that 
each  of  these  expressions  satisfies  the  same  law  of  duality,  as  the 
individual  symbols  which  it  contains.  Thus, 

xyz  x xyz  - xyz ; 

xy  (1  “ z)  x xy  (1  - z)  = xy  (1  - z)  ; 

and  so  on.  Any  such  term  as  the  above  we  shall  designate  as 
a “ class  term,”  because  it  expresses  a class  of  things  by  means 
of  the  common  properties  or  names  of  the  individual  members  of 
such  class. 

Secondly,  If  we  speak  of  a collection  of  things,  different 
portions  of  which  are  defined  by  different  properties,  names,  or 
attributes,  the  expressions  for  those  different  portions  must  be 
separately  formed,  and  then  connected  by  the  sign  + . But  if 
the  collection  of  which  we  desire  to  speak  has  been  formed  by 
excluding  from  some  wider  collection  a defined  portion  of  its 
members,  the  sign  - must  be  prefixed  to  the  symbolical  expres- 
sion of  the  excluded  portion.  Respecting  the  use  of  these  sym- 
bols some  further  observations  may  be  added. 

6.  Speaking  generally,  the  symbol  + is  the  equivalent  of  the 
conjunctions  “ and,”  “or,”  and  the  symbol  -,  the  equivalent  of 
the  preposition  “ except.”  Of  the  conjunctions  “ and”  and  “ or,” 
the  former  is  usually  employed  when  the  collection  to  be  de- 
scribed forms  the  subject,  the  latter  when  it  forms  the  predicate, 
of  a proposition.  “ The  scholar  and  the  man  of  the  world  de- 
sire happiness,”  may  be  taken  as  an  illustration  of  one  of  these 
cases.  “ Things  possessing  utility  are  either  productive  of  plea- 
sure or  preventive  of  pain,”  may  exemplify  the  other.  Now 
whenever  an  expression  involving  these  particles  presents  itself 
in  a primary  proposition,  it  becomes  very  important  to  know 
whether  the  groups  or  classes  separated  in  thought  by  them  are 
intended  to  be  quite  distinct  from  each  other  and  mutually  ex- 
clusive, or  not.  Does  the  expression,  “ Scholars  and  men  of  the 
world,”  include  or  exclude  those  who  are  both  ? Does  the  ex- 


56 


DIVISION  OF  PROPOSITIONS. 


[CHAP.  IV. 

pression,  “ Either  productive  of  pleasure  or  preventive  of  pain,” 
include  or  exclude  things  which  possess  both  these  qualities  ? I 
apprehend  that  in  strictness  of  meaning  the  conjunctions  “and,” 
“ or,”  do  possess  the  power  of  separation  or  exclusion  here  re- 
ferred to ; that  the  formula,  “ All  x's  are  either  y s or  z s,” 
rigorously  interpreted,  means,  “ All  as  are  either  y’s,  but  not  z’s,” 
or,  “ z’s  but  not  y s.”  But  it  must  at  the  same  time  be  admitted, 
that  the  “jus  et  norma  loquendi”  seems  rather  to  favour  an  oppo- 
site interpretation.  The  expression,  “ Either  y s or  2’s,”  would 
generally  be  understood  to  include  things  that  are  y s and  z' s at 
the  same  time,  together  with  tilings  which  come  under  the  one, 
but  not  the  other.  Remembering,  however,  that  the  symbol  + 
does  possess  the  separating  power  ivhich  has  been  the  subject  of 
discussion,  we  must  resolve  any  disjunctive  expression  which  may 
come  before  us  into  elements  really  separated  in  thought,  and 
then  connect  their  respective  expressions  by  the  symbol  + . 

And  thus,  according  to  the  meaning  implied,  the  expression, 
“ Things  which  are  either  x's  or  y s,”  will  have  two  different  sym- 
bolical equivalents.  If  we  mean,  “ Things  which  are  x's,  but 
not  y s,  or  y s,  but  not  x’s,”  the  expression  will  be 

~y)  + yi}  -«); 

the  symbol  x standing  for  x's,  y for  y s.  If,  however,  we  mean, 
“ Things  which  are  either  x's,  or,  if  not  x’s,  then  y’s,”  the  ex- 
pression will  be 

x + y (1  - x). 

This  expression  supposes  the  admissibility  of  things  which  are 
both  x's  and  y's  at  the  same  time.  It  might  more  fully  be  ex- 
pressed in  the  form 

*y  + x (1  - y)  + y (1  - x) ; 

but  this  expression,  on  addition  of  the  two  first  terms,  only  re- 
produces the  former  one. 

Let  it  be  observed  that  the  expressions  above  given  satisfy 
the  fundamental  law  of  duality  (III.  16).  Thus  we  have 

{x  (1  - y)  + y (1  - x))2  = x (1  -y)  + y (1  - x), 

{x  + y (1  - x) }2  = x + y (1  - x). 

It  will  be  seen  hereafter,  that  this  is  but  a particular  manifesta- 


CHAP.  IV.]  DIVISION  OP  PROPOSITIONS.  57 

tion  of  a general  law  of  expressions  representing  “ classes  or 
collections  of  things.” 

7.  The  results  of  these  investigations  may  be  embodied  in 
the  following  rule  of  expression. 

Rule. — Express  simple  names  or  qualities  by  the  symbols  x,  y,  z, 
Sfc.,  their  contraries  by  1-  x,  1 - y,  1 - z,  8fc.;  classes  of  things 
defined  by  common  names  or  qualities , by  connecting  the  correspond- 
ing symbols  as  in  multiplication  ; collections  of  things , consisting  of 
portions  different  from  each  other , by  connecting  the  expressions  of 
those  portions  by  the  sign  + . In  particular , let  the  expression , “ Either 
x’s  or  ys,"  be  expressed  by  x(l-  y)  + y x),  when  the  classes  de- 
noted by  x and  y are  exclusive , by  x + y ( 1 - x)  when  they  are  not 
exclusive.  Similarly  let  the  expression , “ Either  x's,  or  ys,  or  z's,"  be 
expressed  by  x (1  - y)  (1  - z)  + y (1  - x)  (1  - z)  + z (1  - x ) (1  - y), 
when  the  classes  denoted  by  x,  y,  and  z,  are  designed  to  be  mutually 
exclusive,  by  x + y (1  - x)  + z (1  - x)  (1  -y),  when  they  are  not  meant 
to  be  exclusive,  and  so  on. 

8.  On  this  rule  of  expression  is  founded  the  converse  rule  of 
interpretation.  Both  these  will  be  exemplified  with,  perhaps, 
sufficient  fulness  in  the  following  instances.  Omitting  for  bre- 
vity the  universal  subject  “ things,”  or  “ beings,”  let  us  assume 

x = hard,  y = elastic,  z = metals  ; 
and  we  shall  have  the  following  results : 

“ Non-elastic  metals,”  will  be  expressed  by  z (1  - y)  ; 

“ Elastic  substances  with  non-elastic  metals,”  by  y + z (1  - y) ; 

“ Hard  substances,  except  metals,”  by  x -z  ; 

“ Metallic  substances,  except  those  which  are  neither  hard  nor 

elastic,”  by  z -z  (1  - x)  (1  - y),  or  by  2 { 1 - (1  - x)  (1  - y)), 

vide  (6),  Chap.  II. 

In  the  last  example,  what  we  had  really  to  express  was  “ Metals, 
except  not  hard,  not  elastic,  metals.”  Conjunctions  used  be- 
tween adjectives  are  usually  superfluous,  and,  therefore,  must 
not  be  expressed  symbolically. 

Thus,  “ Metals  hard  and  elastic,”  is  equivalent  to  “ Hard 
elastic  metals,”  and  expressed  by  xyz. 

Take  next  the  expression,  “ Hard  substances,  except  those 


58  DIVISION  OF  PROPOSITIONS.  [CHAP.  IV. 

which  are  metallic  and  non-clastic,  and  those  which  are  elastic 
and  non-metallic.”  Here  the  word  those  means  hard  substances, 
so  that  the  expression  really  means,  Hard  substa?ices  except  hard 
substances,  metallic,  non-elastic , and  hard  substances  non-metallic, 
elastic;  the  word  except  extending  to  both  the  classes  which 
follow  it.  The  complete  expression  is 

x - [xz  (1  - y)  + xy  (1  - z)}  ; 
or,  x - xz  (1  - y)  - xy  (1  - z). 

9.  The  preceding  Proposition,  with  the  different  illustrations 
which  have  been  given  of  it,  is  a necessary  preliminary  to  the 
following  one,  which  will  complete  the  design  of  the  present 
chapter. 

Proposition  III. 

To  deduce  from  an  examination  of  their  possible  varieties  a gene- 
ral method  for  the  expression  of  Primary  or  Concrete  Propositions. 

A primary  proposition,  in  the  most  general  sense,  consists  of 
two  terms,  between  which  a relation  is  asserted  to  exist.  These 
terms  are  not  necessarily  single-worded  names,  but  may  represent 
any  collection  of  objects,  such  as  we  have  been  engaged  in  consi- 
dering in  the  previous  sections.  The  mode  of  expressing  those 
terms  is,  therefore,  comprehended  in  the  general  precepts  above 
given,  and  it  only  remains  to  discover  how  the  relations  between 
the  terms  are  to  be  expressed.  This  will  evidently  depend  upon 
the  nature  of  the  relation,  and  more  particularly  upon  the  ques- 
tion whether,  in  that  relation,  the  terms  are  understood  to  be 
universal  or  particular,  i.  e.  whether  we  speak  of  the  whole  of 
that  collection  of  objects  to  which  a term  refers,  or  indefinitely  of 
the  whole  or  of  a part  of  it,  the  usual  signification  of  the  prefix, 
“ some.” 

Suppose  that  we  wish  to  express  a relation  of  identity  be- 
tween the  two  classes,  “ Fixed  Stars”  and  “ Suns,”  i.  e.  to 
express  that  “ All  fixed  stars  are  suns,”  and  “ All  suns  are  fixed 
stars.”  Here,  if  x stand  for  fixed  stars,  and  y for  suns,  we  shall 
have 

x = y 

for  the  equation  required. 


CHAP.  IV.]  DIVISION  OF  PROPOSITIONS.  59 

In  the  proposition,  “ All  fixed  stars  are  suns,”  the  term  “all 
fixed  stars”  would  be  called  the  subject , and  “ suns”  the  predi- 
cate. Suppose  that  Ave  extend  the  meaning  of  the  terms  subject 
and  predicate  in  the  following  manner.  By  subject  let  us  mean 
the  first  term  of  any  affirmative  proposition,  i.  e.  the  term  Avhich 
precedes  the  copula  is  or  are  ; and  by  predicate  let  us  agree  to 
mean  the  second  term,  i.  e.  the  one  which  follows  the  copula ; 
and  let  us  admit  the  assumption  that  either  of  these  may  be  uni- 
versal or  particular,  so  that,  in  either  case,  the  Avhole  class  may 
be  implied,  or  only  a part  of  it.  Then  >ve  shall  have  the  folio av- 
ing  Rule  for  cases  such  as  the  one  in  the  last  example: — 

10.  Rule. — When  both  Subject  and  Predicate  of  a Proposition 
are  universal, form  the  separate  expressions  for  them,  and  connect  them 
by  the  sign  =. 

This  case  will  usually  present  itself  in  the  expression  of,  the 
definitions  of  science,  or  of  subjects  treated  after  the  manner  of 
pure  science.  Mr.  Senior’s  definition  of  Avealth  affords  a good 
example  of  this  kind,  viz. : 

“ Wealth  consists  of  things  transferable,  limited  in  supply, 
and  either  productive  of  pleasure  or  preventive  of  pain.” 

Before  proceeding  to  express  this  definition  symbolically,  it 
must  be  remarked  that  the  conjunction  and  is  superfluous. 
Wealth  is  really  defined  by  its  possession  of  three  properties  or 
qualities,  not  by  its  composition  out  of  three  classes  or  collections 
of  objects.  Omitting  then  the  conjunction  and , let  us  make 

w = wealth. 

t - things  transferable. 
s = limited  in  supply. 
p = productive  of  pleasure. 
r = preventive  of  pain. 

Now  it  is  plain  from  the  nature  of  the  subject,  that  the  ex- 
pression, “ Either  productive  of  pleasure  or  preventive  of  pain,” 
in  the  above  definition,  is  meant  to  be  equivalent  to  “ Either  pro- 
ductive of  pleasure ; or,  if  not  productive  of  pleasure,  preventive 
of  pain.”  Thus  the  class  of  things  Avhich  the  above  expression, 
taken  alone,  would  define,  would  consist  of  all  things  productive 


60 


DIVISION  OF  PROPOSITIONS. 


[CHAP.  IV. 

of  pleasure,  together  with  all  things  not  productive  of  pleasure, 
but  preventive  of  pain,  and  its  symbolical  expression  would  be 

p + ( 1 - p)  r. 

If  then  we  attach  to  this  expression  placed  in  brackets  to  denote 
that  both  its  terms  are  referred  to,  the  symbols  s and  t limiting 
its  application  to  things  “transferable”  and  “limited  in  supply,” 
we  obtain  the  folloAving  symbolical  equivalent  for  the  original 
definition,  viz. : 

w = st  [p  + r (1  - p)}.  (1) 

If  the  expression,  “ Either  productive  of  pleasure  or  preventive  of 
pain,”  were  intended  to  point  out  merely  those  things  which  are 
productive  of  pleasure  without  being  preventive  of  pain,  p (1  - r), 
or  preventive  of  pain,  without  being  productive  of  pleasure, 
r (1  - p)  (exclusion  being  made  of  those  things  which  are  both 
productive  of  pleasure  and  preventive  of  pain),  the  expression  in 
symbols  of  the  definition  would  be 

w - st  [p  (1  - r)  + r (1  -/>)}.  (2) 

All  this  agrees  with  what  has  before  been  more  generally  stated. 

The  reader  may  be  curious  to  inquire  what  effect  would  be 
produced  if  we  literally  translated  the  expression,  “ Things  pro- 
ductive of  pleasure  or  preventive  of  pain,”  by  p + r,  making  the 
symbolical  equation  of  the  definition  to  be 

w = st  (p  + r ).  (3) 

The  answer  is,  that  this  expression  would  be  equivalent  to  (2), 
with  the  additional  implication  that  the  classes  of  things  denoted 
by  stp  and  sir  are  quite  distinct,  so  that  of  things  transferable 
and  limited  in  supply  there  exist  none  in  the  universe  which  are 
at  the  same  time  both  productive  of  pleasure  and  preventive  of 
pain.  How  the  full  import  of  any  equation  may  be  determined 
will  be  explained  hereafter.  What  has  been  said  may  show  that  be- 
fore attempting  to  translate  our  data  into  the  rigorous  language 
of  symbols,  it  is  above  all  things  necessary  to  ascertain  the  in- 
tended. import  of  the  words  we  are  using.  But  this  necessity 
cannot  be  regarded  as  an  evil  by  those  who  value  correctness  of 


CHAP.  IV.] 


DIVISION  OF  PROPOSITIONS. 


61 


thought,  and  regard  the  right  employment  of  language  as  both 
its  instrument  and  its  safeguard. 

1 1 . Let  us  consider  next  the  case  in  which  the  predicate  of 
the  proposition  is  particular,  e.  g.  “ All  men  are  mortal.” 

In  this  case  it  is  clear  that  our  meaning  is,  “ All  men  are 
some  mortal  beings,”  and  we  must  seek  the  expression  of  the 
predicate,  “ some  mortal  beings.”  Represent  then  by  v,  a class 
indefinite  in  every  respect  but  this,  viz.,  that  some  of  its  members 
are  mortal  beings,  and  let  x stand  for  “mortal  beings,” then  will 
vx  represent  “ some  mortal  beings.”  Hence  if  y represent  men, 
the  equation  sought  will  be 

y - vx. 

From  such  considerations  we  derive  the  following  Rule,  for 
expressing  an  affirmative  universal  proposition  whose  predicate 
is  particular : 

Rule. — Express  as  before  the  subject  and  the  predicate , attach 
to  the  latter  the  indefinite  symbol  v,  and  equate  the  expressions. 

It  is  obvious  that  v is  a symbol  of  the  same  kind  as  x,  y,  &c., 
and  that  it  is  subject  to  the  general  law, 

v 2 = v,  or  v (1  - v)  ~ 0. 

Thus,  to  express  the  proposition,  “ The  planets  are  either 
primary  or  secondary,”  we  should,  according  to  the  rule,  proceed 
thus : 

Let  x represent  planets  (the  subject)  ; 
y = primary  bodies  ; 
z = secondary  bodies ; 

then,  assuming  the  conjunction  “or”  to  separate  absolutely  the 
class  of  “primary”  from  that  of  “ secondary”  bodies,  so  far  as 
they  enter  into  our  consideration  in  the  proposition  given,  we 
find  for  the  equation  of  the  proposition 

x = v [y{\  - z)  + z{\  -y)}.  (4) 

It  may  be  worth  while  to  notice,  that  in  this  case  the  literal 
translation  of  the  premises  into  the  form 

x - v (y  + z) 


(5) 


DIVISION  OF  PROPOSITIONS. 


62 


[chap.  IV. 


would  be  exactly  equivalent,  v being  an  indefinite  class  6yrabol. 
The  form  (4)  is,  however,  the  better,  as  the  expression 

y (1  -z)  + z{\  -y) 

consists  of  terms  representing  classes  quite  distinct  from  each 
other,  and  satisfies  the  fundamental  law  of  duality. 

If  we  take  the  proposition,  “ The  heavenly  bodies  are  either 
suns,  or  planets,  or  comets,”  representing  these  classes  of  things 
by  w,  x , y,  z , respectively,  its  expression,  on  the  supposition  that 
none  of  the  heavenly  bodies  belong  at  once  to  two  of  the  divi- 
sions above  mentioned,  will  be 

w = v{x(l  - y)  (1  - z)  + y(l  - x)  (1  - z)  + z (l  -x)  (1  -y)). 

If,  however,  it  were  meant  to  be  implied  that  the  heavenly 
bodies  were  either  suns,  or,  if  not  suns,  planets,  or,  if  neither  suns 
nor  planets,  fixed  stars,  a meaning  which  does  not  exclude  the 
supposition  of  some  of  them  belonging  at  once  to  two  or  to  all 
three  of  the  divisions  of  suns,  planets,  and  fixed  stars, — the  ex- 
pression required  would  be 

w = v [x  + y - x)  + z ( \ - x)  {\  - y)).  (6) 

The  above  examples  belong  to  the  class  of  descriptions,  not 
definitions.  Indeed  the  predicates  of  propositions  are  usually 
particular.  When  this  is  not  the  case,  either  the  predicate  is  a 
singular  term,  or  we  employ,  instead  of  the  copula  “ is”  or  “ are,” 
some  form  of  connexion,  which  implies  that  the  predicate  is  to  be 
taken  universally. 

12.  Consider  next  the  case  of  universal  negative  propositions, 
e.  g.  “ No  men  are  perfect  beings.” 

Now  it  is  manifest  that  in  this  case  we  do  not  speak  of  a class 
termed  “no  men,”  and  assert  of  this  class  that  all  its  members 
are  “ perfect  beings.”  But  we  virtually  make  an  assertion  about 
“ all  men ” to  the  effect  that  they  are  “ not  perfect  beings .”  Thus 
the  true  meaning  of  the  proposition  is  this  : 

“ All  men  (subject)  are  (copula)  not  perfect  (predicate)  ;” 
whence,  if  y represent  “ men,”  and  x “ perfect  beings,”  we  shall 
have 


y = «0  - x ), 


DIVISION  OF  PROPOSITIONS. 


63 


CHAP.  IV.] 

and  similarly  in  any  other  case.  Thus  we  have  the  following 
Rule : 

Rule. — To  express  any  proposition  of  the  form  “ No  x's  are 
y s,"  convert  it  into  the  form  “ All  x's  are  not  y's,"  and  then  proceed 
as  in  the  previous  case. 

13.  Consider,  lastly,  the  case  in  which  the  subject  of  the 
proposition  is  particular,  e.  g.  “ Some  men  are  not  wise.”  Here, 
as  has  been  remarked,  the  negative  not  may  properly  be  referred, 
certainly,  at  least,  for  the  ends  of  Logic,  to  the  predicate  wise  ; 
for  we  do  not  mean  to  say  that  it  is  not  true  that  “ Some  men 
are  wise,”  but  we  intend  to  predicate  of  “ some  men”  a want  of 
wisdom.  The  requisite  form  of  the  given  proposition  is,  there- 
fore, “ Some  men  are  not-wise.”  Putting,  then,  y for  “men,” 
x for  “ wise,”  i.  e.  “ wise  beings,”  and  introducing  v as  the  sym- 
bol of  a class  indefinite  in  all  respects  but  this,  that  it  contains 
some  individuals  of  the  class  to  whose  expression  it  is  prefixed, 
we  have 

vy  = v (1  - x). 

14.  We  may  comprise  all  that  we  have  determined  in  the 
following  general  Rule : 

GENERAL  RULE  FOR  THE  SYMBOLICAL  EXPRESSION  OF  PRIMARY 
PROPOSITIONS. 

1st.  If  the  proposition  is  affirmative,  form  the  expression  of  the 
subject  and  that  of  the  predicate.  Should  either  of  them  be  particular, 
attach  to  it  the  indefinite  symbol  v,  and  then  equate  the  resulting  ex- 
pressions. 

2ndly.  If  the  proposition  is  negative,  express  first  its  true  mean- 
ing by  attaching  the  negative  particle  to  the  predicate,  then  proceed  as 
above. 

One  or  two  additional  examples  may  suffice  for  illustration. 

Ex. — “ No  men  are  placed  in  exalted  stations,  and  free  from 
envious  regards.” 

Let  y represent  “ men,”  x,  “ placed  in  exalted  stations,”  z, 
“ free  from  envious  regards.” 

Now  the  expression  of  the  class  described  as  “placed  in 


64 


DIVISION  OF  PROPOSITIONS. 


[CHAP.  IV. 


exalted  station,”  and  “ free  from  envious  regards,”  is  xz.  Hence 
the  contrary  class,  i.  e.  they  to  whom  this  description  does  not 
apply,  will  be  represented  by  1 - xz,  and  to  this  class  all  men 
are  referred.  Hence  we  have 

y = v(  1 - xz). 

If  the  proposition  thus  expressed  had  been  placed  in  the  equiva- 
lent form,  “ Men  in  exalted  stations  are  not  free  from  envious 
regards,”  its  expression  would  have  been 

yx  = v (1  - z). 

It  will  hereafter  appear  that  this  expression  is  really  equivalent 
to  the  previous  one,  on  the  particular  hypothesis  involved,  viz., 
that  v is  an  indefinite  class  symbol. 

Ex. — “ No  men  are  heroes  but  those  who  unite  self-denial  to 
courage.” 

Let  x = “ men,”  y = “ heroes,”  z = “ those  who  practise  self- 
denial,”  w,  “ those  who  possess  courage.” 

The  assertion  really  is,  that  “ men  who  do  not  possess  cou- 
rage and  practise  self-denial  are  not  heroes.” 

Hence  we  have 

x ( 1 - zw ) = v ( 1 - y) 
for  the  equation  required. 

15.  Inclosing  this  Chapter  it  may  be  interesting  to  compare 
together  the  great  leading  types  of  propositions  symbolically  ex- 
pressed. If  we  agree  to  represent  by  X and  Y the  symbolical 
expressions  of  the  “terms,”  or  things  related,  those  types  will 
be 

X =vY, 

X = Y, 
vX  =vY. 


In  the  first,  the  predicate  only  is  particular ; in  the  second,  both 
terms  are  universal ; in  the  third,  both  are  particular.  Some  mi- 
nor forms  are  really  included  under  these.  Thus,  if  Y = 0,  the 
second  form  becomes 

X=0; 

and  if  Y = 1 it  becomes 

X - \ ; 


CHAP.  IV.] 


DIVISION  OF  PROPOSITIONS. 


65 


both  which  forms  admit  of  interpretation.  It  is  further  to  be 
noticed,  that  the  expressions  X and  Y,  if  founded  upon  a suffi- 
ciently careful  analysis  of  the  meaning  of  the  “ terms”  of  the 
proposition,  will  satisfy  the  fundamental  law  of  duality  which 
requires  that  we  have 

P=Ior  X(1  - X)  = 0, 

Y2  = Y or  Y(1  - Y)  = 0. 


G(j 


PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 


CHAPTER  V. 

OF  THE  FUNDAMENTAL  PRINCIPLES  OF  SYMBOLICAL  REASONING,  AND 
OF  THE  EXPANSION  OR  DEVELOPMENT  OF  EXPRESSIONS  INVOLV- 
ING LOGICAL  SYMBOLS. 

1 . r I ''HE  previous  chapters  of  this  work  have  been  devoted  to 
the  investigation  of  the  fundamental  laws  of  the  opera- 
tions of  the  mind  in  reasoning;  of  their  development  in  the 
laws  of  the  symbols  of  Logic ; and  of  the  principles  of  expression, 
by  which  that  species  of  propositions  called  primary  may  be  repre- 
sented in  the  language  of  symbols.  These  inquiries  have  been 
in  the  strictest  sense  preliminary.  They  form  an  indispensable 
introduction  to  one  of  the  chief  objects  of  this  treatise — the  con- 
struction of  a system  or  method  of  Logic  upon  the  basis  of  an 
exact  summary  of  the  fundamental  laws  of  thought.  There  are 
certain  considerations  touching  the  nature  of  this  end,  and  the 
means  of  its  attainment,  to  which  I deem  it  necessary  here  to 
direct  attention. 

2.  I would  remark  in  the  first  place  that  the  generality  of  a 
method  in  Logic  must  very  much  depend  upon  the  generality  of 
its  elementary  processes  and  laws.  We  have,  for  instance,  in  the 
previous  sections  of  this  work  investigated,  among  other  things, 
the  laws  of  that  logical  process  of  addition  which  is  symbolized 
by  the  sign  +.  Now  those  laws  have  been  determined  from  the 
study  of  instances,  in  all  of  which  it  has  been  a necessary  condi- 
tion, that  the  classes  or  things  added  together  in  thought  should 
be  mutually  exclusive.  The  expression  x + y seems  indeed  un- 
interpretable, unless  it  be  assumed  that  the  things  represented 
by  x and  the  things  represented  by  y are  entirely  separate  ; 
that  they  embrace  no  individuals  in  common.  And  conditions 
analogous  to  this  have  been  involved  in  those  acts  of  conception 
from  the  study  of  which  the  laws  of  the  other  symbolical  opera- 
tions have  been  ascertained.  The  question  then  arises,  whether 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING.  67 

it  is  necessary  to  restrict  the  application  of  these  symbolical  laws 
and  processes  by  the  same  conditions  of  interpretability  under 
which  the  knowledge  of  them  was  obtained.  If  such  restriction 
is  necessary,  it  is  manifest  that  no  such  thing  as  a general 
method  in  Logic  is  possible.  On  the  other  hand,  if  such  restric- 
tion is  unnecessary,  in  what  light  are  we  to  contemplate  pro- 
cesses which  appear  to  be  uninterpretable  in  that  sphere  of  thought 
which  they  are  designed  to  aid  ? These  questions  do  not  belong 
to  the  science  of  Logic  alone.  They  are  equally  pertinent  to  every 
developed  form  of  human  reasoning  which  is  based  upon  the 
employment  of  a symbolical  language. 

3.  I would  observe  in  the  second  place,  that  this  apparent 
failure  of  correspondency  between  process  and  interpretation  does 
not  manifest  itself  in  the  ordinary  applications  of  human  rea- 
son. For  no  operations  are  there  performed  of  which  the  mean- 
ing and  the  application  are  not  seen ; and  to  most  minds  it  does 
not  suffice  that  merely  formal  reasoning  should  connect  their 
premises  and  their  conclusions ; but  every  step  of  the  connecting 
train,  every  mediate  result  which  is  established  in  the  course  of 
demonstration,  must  be  intelligible  also.  And  without  doubt, 
this  is  both  an  actual  condition  and  an  important  safeguard,  in 
the  reasonings  and  discourses  of  common  life. 

There  are  perhaps  many  who  would  be  disposed  to  extend 
the  same  principle  to  the  general  use  of  symbolical  language  as 
an  instrument  of  reasoning.  It  might  be  argued,  that  as  the 
laws  or  axioms  which  govern  the  use  of  symbols  are  established 
upon  an  investigation  of  those  cases  only  in  which  interpretation 
is  possible,  we  have  no  right  to  extend  their  application  to  other 
cases  in  which  interpretation  is  impossible  or  doubtful,  even 
though  (as  should  be  admitted)  such  application  is  employed  in 
the  intermediate  steps  of  demonstration  only.  Were  this  ob- 
jection conclusive,  it  must  be  acknowledged  that  slight  ad- 
vantage would  accrue  from  the  use  of  a symbolical  method  in 
Logic.  Perhaps  that  advantage  would  be  confined  to  the  mecha- 
nical gain  of  employing  short  and  convenient  symbols  in  the 
place  of  more  cumbrous  ones.  But  the  objection  itself  is  falla- 
cious. Whatever  our  a -priori  anticipations  might  be,  it  is  an 
unquestionable  fact  that  the  validity  of  a conclusion  arrived  at 


68  PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 

by  any  symbolical  process  of  reasoning,  does  not  depend  upon 
our  ability  to  interpret  the  formal  results  which  have  presented 
themselves  in  the  different  stages  of  the  investigation.  There 
exist,  in  fact,  certain  general  principles  relating  to  the  use  of 
symbolical  methods,  which,  as  pertaining  to  the  particular  sub- 
ject of  Logic,  I shall  first  state,  and  I shall  then  offer  some  re- 
marks upon  the  nature  and  upon  the  grounds  of  their  claim  to 
acceptance. 

4.  The  conditions  of  valid  reasoning,  by  the  aid  of  symbols, 
are — 

1st,  That  a fixed  interpretation  be  assigned  to  the  symbols 
employed  in  the  expression  of  the  data ; and  that  the  laws  of  the 
combination  of  those  symbols  be  correctly  determined  from  that 
interpretation . 

2nd,  That  the  formal  processes  of  solution  or  demonstration 
be  conducted  throughout  in  obedience  to  all  the  laws  deter- 
mined as  above,  without  regard  to  the  question  of  the  interpreta- 
bility  of  the  particular  results  obtained. 

3rd,  That  the  final  result  be  interpretable  in  form,  and  that 
it  be  actually  interpreted  in  accordance  with  that  system  of  in- 
terpretation which  has  been  employed  in  the  expression  of  the 
data.  Concerning  these  principles,  the  following  observations 
may  be  made. 

5.  The  necessity  of  a fixed  interpretation  of  the  symbols  has 
already  been  sufficiently  dwelt  upon  (II.  3).  The  necessity  that 
the  fixed  result  should  be  in  such  a form  as  to  admit  of  that  in- 
terpretation being  applied,  is  founded  on  the  obvious  principle, 
that  the  use  of  symbols  is  a means  towards  an  end,  that  end 
being  the  knowledge  of  some  intelligible  fact  or  truth.  And 
that  this  end  may  be  attained,  the  final  result  which  expresses 
the  symbolical  conclusion  must  be  in  an  interpretable  form.  It 
is,  however,  in  connexion  with  the  second  of  the  above  general 
principles  or  conditions  (V.  4),  that  the  greatest  difficulty  is 
likely  to  be  felt,  and  upon  this  point  a few  additional  words  are 
necessary. 

I would  then  remark,  that  the  principle  in  question  may  be 
considered  as  resting  upon  a general  law  of  the  mind,  the  know- 
ledge of  which  is  not  given  to  us  a priori,  i.  e.  antecedently  to 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING. 


69 


experience,  but  is  derived,  like  the  knowledge  of  the  other  laws 
of  the  mind,  from  the  clear  manifestation  of  the  general  principle 
in  the  particular  instance.  A single  example  of  reasoning,  in 
which  symbols  are  employed  in  obedience  to  laws  founded  upon 
their  interpretation,  but  without  any  sustained  reference  to  that 
interpretation,  the  chain  of  demonstration  conducting  us  through 
intermediate  steps  which  are  not  interpretable,  to  a final  result 
which  is  interpretable,  seems  not  only  to  establish  the  validity  of 
the  particular  application,  but  to  make  known  to  us  the  general 
law  manifested  therein.  No  accumulation  of  instances  can  pro- 
perly add  weight  to  such  evidence.  It  may  furnish  us  with  clearer 
conceptions  of  that  common  element  of  truth  upon  which  the  ap- 
plication of  the  principle  depends,  and  so  prepare  the  way  for  its 
reception.  It  may,  where  the  immediate  force  of  the  evidence  is 
not  felt,  serve  as  a verification,  a posteriori,  of  the  practical  vali- 
dity of  the  principle  in  question.  But  this  does  not  affect  the  posi- 
tion affirmed,  viz.,  that  the  general  principle  must  be  seen  in  the 
particular  instance, — seen  to  be  general  in  application  as  well  as 
true  in  the  special  example.  The  employment  of  the  uninterpre- 
table symbol  ^/  - 1 , in  the  intermediate  processes  of  trigonometry, 
furnishes  an  illustration  of  what  has  been  said.  I apprehend  that 
there  is  no  mode  of  explaining  that  application  which  does  not 
covertly  assume  the  very  principle  in  question.  But  that  prin- 
ciple, though  not,  as  I conceive,  warranted  by  formal  reasoning 
based  upon  other  grounds,  seems  to  deserve  a place  among  those 
axiomatic  truths  which  constitute,  in  some  sense,  the  foundation 
of  the  possibility  of  general  knowledge,  and  which  may  properly 
be  regarded  as  expressions  of  the  mind’s  own  laws  and  consti- 
tution. 

6.  The  following  is  the  mode  in  which  the  principle  above 
stated  will  be  applied  in  the  present  work.  It  has  been  seen, 
that  any  system  of  propositions  may  be  expressed  by  equations 
involving  symbols  x,  y,  z,  which,  whenever  interpretation  is  pos- 
sible, are  subject  to  laws  identical  in  form  with  the  laws  of  a sys- 
tem of  quantitative  symbols,  susceptible  only  of  the  values  0 and 
1 (II.  15).  But  as  the  formal  processes  of  reasoning  depend  only 
upon  the  laws  of  the  symbols,  and  not  upon  the  nature  of  their 
interpretation,  we  are  permitted  to  treat  the  above  symbols, 


70  PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 

x,  y,  z,  as  if  they  were  quantitative  symbols  of  the  kind  above 
described.  We  may  in  fact  lay  aside  the  logical  interpretation  of 
the  symbols  in  the  given  equation ; convert  them  into  quantitative  sym- 
bols, susceptible  only  of  the  values  0 and  1 ; perform  upon  them  as  such 
all  the  requisite  processes  of  solution;  and  finally  restore  to  them  their 
logical  interpretation.  And  this  is  the  mode  of  procedure  which 
will  actually  be  adopted,  though  it  will  be  deemed  unnecessary 
to  restate  in  every  instance  the  nature  of  the  transformation  em- 
ployed. The  processes  to  which  the  symbols  x,  y,  z,  regarded 
as  quantitative  and  of  the  species  above  described,  are  subject,  are 
not  limited  by  those  conditions  of  thought  to  which  they  would, 
if  performed  upon  purely  logical  symbols,  be  subject,  and  a free- 
dom of  operation  is  given  to  us  in  the  use  of  them,  without 
which,  the  inquiry  after  a general  method  in  Logic  would  be  a 
hopeless  quest. 

Now  the  above  system  of  processes  would  conduct  us  to  no 
intelligible  result,  unless  the  final  equations  resulting  therefrom 
were  in  a form  which  should  render  their  interpretation,  after 
restoring  to  the  symbols  their  logical  significance,  possible. 
There  exists,  however,  a general  method  of  reducing  equations 
to  such  a form,  and  the  remainder  of  this  chapter  will  be  devoted 
to  its  consideration.  I shall  say  little  concerning  the  way  in 
which  the  method  renders  interpretation  possible, — this  point 
being  reserved  for  the  next  chapter, — but  shall  chiefly  confine 
myself  here  to  the  mere  process  employed,  which  may  be  cha- 
racterized as  a process  of  “ development.”  As  introductory  to 
the  nature  of  this  process,  it  may  be  proper  first  to  make  a few 
observations. 

7.  Suppose  that  we  are  considering  any  class  of  things  with 
reference  to  this  question,  viz.,  the  relation  in  which  its  members 
stand  as  to  the  possession  or  the  want  of  a certain  property  x.  As 
every  individual  in  the  proposed  class  either  possesses  or  does 
not  possess  the  property  in  question,  we  may  divide  the  class 
into  two  portions,  the  former  consisting  of  those  individuals 
which  possess,  the  latter  of  those  which  do  not  possess,  the  pro- 
perty. This  possibility  of  dividing  in  thought  the  whole  class 
into  two  constituent  portions,  is  antecedent  to  all  knowledge  of 
the  constitution  of  the  class  derived  from  any  other  source ; of 


71 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING. 


which  knowledge  the  effect  can  only  be  to  inform  us,  more  or 
less  precisely,  to  what  further  conditions  the  portions  of  the  class 
which  possess  and  which  do  not  possess  the  given  property  are 
subject.  Suppose,  then,  such  knowledge  is  to  the  following  effect, 
viz.,  that  the  members  of  that  portion  which  possess  the  property 
x,  possess  also  a certain  property  u,  and  that  these  conditions 
united  are  a sufficient  definition  of  them.  We  may  then  repre- 
sent that  portion  of  the  original  class  by  the  expression  ux  (II.  6). 
If,  further,  we  obtain  information  that  the  members  of  the  ori- 
ginal class  which  do  not  possess  the  property  x,  are  subject  to  a 
condition  v,  and  are  thus  defined,  it  is  clear,  that  those  members 
■will  be  represented  by  the  expression  v (1  -x).  Hence  the  class 
in  its  totality  will  be  represented  by 

ux  + v (1  - x)  ; 


which  may  be  considered  as  a general  developed  form  for  the 
expression  of  any  class  of  objects  considered  Avith  reference  to 
the  possession  or  the  want  of  a given  property  x. 

The  general  form  thus  established  upon  purely  logical 
grounds  may  also  be  deduced  from  distinct  considerations  of 
formal  laiv,  applicable  to  the  symbols  x,  y,  z,  equally  in  their 
logical  and  in  their  quantitative  interpretation  already  referred  to 
(V.6). 

8.  Definition. — Any  algebraic  expression  involving  a sym- 
bol x is  termed  a function  of  x,  and  may  be  represented  under 
the  abbreviated  general  form  f(x).  Any  expression  involving 
two  symbols,  x and  y,  is  similarly  termed  a function  of  x and  y, 
and  may  be  represented  under  the  general  form  f(x,  y),  and  so 
on  for  any  other  case. 

Thus  the  form  f (x)  would  indifferently  represent  any  of  the 


following  functions,  viz.,  x,  1 - x, 


1 + x 


- — -,  &c. ; and  fix,  y)  Avould 

equally  represent  any  of  the  forms  x + y,  x -2 y,  - — &c. 

x — ly 

On  the  same  principles  of  notation,  if  in  any  function  f(x), 
Ave  change  x into  1,  the  result  Avill  be  expressed  by  the  form 
/( 1)  ; if  in  the  same  function  Ave  change  x into  0,  the  result  Avill 
be  expressed  by  the  form  /( 0).  Thus,  if  f (x)  represent  the 


72 


PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 


function 


a +x 
a - 2x 


/( 1)  will  represent 


a + 1 
a - 2 ’ 


and  / (0)  will  repre- 


, a 
sent  - . 
a 

9.  Definition. — Any  function  f(x),  in  which  # is  a logical 
symbol,  or  a symbol  of  quantity  susceptible  only  of  the  values 
0 and  1,  is  said  to  be  developed,  when  it  is  reduced  to  the  form 
ax  + b (1  - x),  a and  b being  so  determined  as  to  make  the  result 
equivalent  to  the  function  from  which  it  was  derived. 

This  definition  assumes,  that  it  is  possible  to  represent  any 
function  f (x)  in  the  form  supposed.  The  assumption  is  vindi- 
cated in  the  following  Proposition. 


Proposition  I. 


10.  To  develop  any  function  f (x)  in  which  x is  a logical  symbol. 

By  the  principle  which  has  been  asserted  in  this  chapter,  it 
is  lawful  to  treat  a as  a quantitative  symbol,  susceptible  only  of 
the  values  0 and  1. 

Assume  then, 

f(x)  = ax  + b ( 1 - x), 
and  making  x = 1,  we  have 

/(!)  = «• 

Again,  in  the  same  equation  making  x = 0,  we  have 

m = &■ 

Hence  the  values  of  a and  b are  determined,  and  substituting 
them  in  the  first  equation,  we  have 

/(*W(l)*+/(0)(l-«);  (1) 

as  the  development  sought.*  The  second  member  of  the  equa- 


* To  some  it  may  be  interesting  to  remark,  that  the  development  of  /Or) 
obtained  in  this  chapter,  strictly  holds,  in  the  logical  system,  tl^e  place  of  the 
expansion  of  f (x)  in  ascending  powers  of  x in  the  system  of  ordinary  algebra. 
Thus  it  may  be  obtained  by  introducing  into  the  expression  of  Taylor’s  well- 
known  theorem,  viz. : 

/ (*)  =/  (0)  + / (0)  X +f"  (0)  A-  4 (0)  jfL-,  &C. 

the  condition  x (1  - x)  = 0,  whence  we  find  x*  = x,  x3  — x,  &c.,  and 


(1) 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING. 


73 


tion  adequately  represents  the  function  f (x),  whatever  the  form 
of  that  function  may  be.  For  a;  regarded  as  a quantitative  sym- 
bol admits  only  of  the  values  0 and  1,  and  for  each  of  these 
values  the  development 

/ (1)  x +f  (0)  (I-*), 

assumes  the  same  value  as  the  function/ (x). 

As  an  illustration,  let  it  be  required  to  develop  the  function 

l Q 

- — — . Here,  when  x = 1,  we  find  /( 1)  = - , and  when  x = 0, 
1 + 2x  o 

we  find/(0)  = y , or  1.  Hence  the  expression  required  is 

l+x  2 

t72i-3*+l-*i 

and  this  equation  is  satisfied  for  each  of  the  values  of  which  the 
symbol  x is  susceptible. 


Proposition  II. 

To  expand  or  develop  a function  involving  any  number  of  logical 
symbols. 

Let  us  begin  with  the  case  in  which  there  are  two  symbols, 
x and  y,  and  let  us  represent  the  function  to  be  developed  by 

First,  considering  / (x,  y)  as  a function  of  x alone,  and  ex- 
panding it  by  the  general  theorem  (1),  we  have 

f(*>y)tsf(i>y)*-+f(0,y)(i-x)i  (2) 


/(*)  =/(0)  + {/  (0)  +45  + + &C. } X.  (2) 

But  making  in  (1),  x = 1,  we  get 

/(i)  =/(  o)  +/  (0)  +4?  + rS + &c- ; 

whence 

/'  (0)  +45 + &c-  =^(i) 

and  (2)  becomes,  on  substitution, 

/(x)=/(0)  + {/(l)-/(0)}x, 

= /(l)x  +/ (0)  (1  - x), 

the  form  in  question.  This  demonstration  in  supposing/ (x)  to  be  developable  in 
a series  of  ascending  powers  of  x is  less  general  than  the  one  in  the  text. 


74 


PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 

wherein /(l,  y)  represents  what  the  proposed  function  becomes, 
when  in  it  for  x we  write  1,  and  / (0,  y)  what  the  said  function 
becomes,  when  in  it  for  x we  write  0. 

Now,  taking  the  coefficient  / (1,  y),  and  regarding  it  as  a func- 
tion of  y,  and  expanding  it  accordingly,  we  have 


/(i,  y)  =/(l,  l)y+/(l,0)(l-y),  (3) 


wherein  /( 1,  1)  represents  what/(l,  y)  becomes  when  y is  made 
equal  to  1,  and  /(l,  0)  what  f(\,y ) becomes  when  y is  made 
equal  to  0. 

In  like  manner,  the  coefficient/ (0,  y ) gives  by  expansion, 

/(0)  y)  =/(0,  1)  y +/(0,  0)  (1  - y).  (4) 

Substitute  in  (2)  for  /( 1,  y),  /( 0,  y),  their  values  given  in  (3) 
and  (4),  and  we  have 

/0>  y)  =/(!>  1 )*y  +/(1,  0)a;(l  -y)  +/( 0,  1)  (1-x )y 

+/(0,  0)(l-*)(l-y),  (5) 

for  the  expansion  required.  Here  /( 1,  1)  represents  what  f(x,y) 
becomes  when  we  make  therein  x = 1,  y = 1 ; /( 1,  0)  represents 
what  f ( x , y)  becomes  when  we  make  therein  x = 1,  y = 0,  and 
so  on  for  the  rest. 


1 - x 


Thus,  if / ( x , y)  represent  the  function  - — -,  we  find 
/0.0-jj.  /(M)-J.o,  /(0, 1) ■=  g,  /(0,0)  = I, 


whence  the  expansion  of  the  given  function  is 


W + 0#  (1  - y)  + J (1  - x)  y + (1  - x)  (1  - y). 

It  will  in  the  next  chapter  be  seen  that  the  forms  and  the 

former  of  which  is  known  to  mathematicians  as  the  symbol  of  in- 
determinate quantity,  admit,  in  such  expressions  as  the  above,  of 
a very  important  logical  interpretation. 

Suppose,  in  the  next  place,  that  we  have  three  symbols  in 
the  function  to  be  expanded,  which  we  may  represent  under  the 
general  form  f(x,  y,  z ).  Proceeding  as  before,  we  get 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING.  75 

f(z,y,  z)=f(\,\,\)xyz+f(\,\,0)xy(\-z)+f(}>  0,  l)o;(l -y)z 
+ /(!»  0)  x (1  - y)  (1  - z)  + /( 0,  1,  1)  (1  - x)  yz 

+ /(0,  1,  0)  (1  - x)y  (1-z)  +/(0,  0,  1)  (1  -x)  (1  -y)z 
+ / ( 0,  0,  0)  (1  -x)  (1  -y)  (1-z), 

in  which  /(l,  1,1)  represents  what  the  function  fix,  y,  z)  be- 
comes when  we  make  therein  x = 1,  y = 1,  z = 1,  and  so  on  for 
the  rest. 

11.  It  is  now  easy  to  see  the  general  law  which  determines 
the  expansion  of  any  proposed  function,  and  to  reduce  the  me- 
thod of  effecting  the  expansion  to  a rule.  But  before  proceeding 
to  the  expression  of  such  a rule,  it  will  be  convenient  to  premise 
the  following  observations  : — 

Each  form  of  expansion  that  we  have  obtained  consists  of  cer- 
tain terms,  into  which  the  symbols  x,  y,  &c.  enter,  multiplied  by 
coefficients,  into  which  those  symbols  do  not  enter.  Thus  the 
expansion  of  f(x)  consists  of  two  terms,  x and  1 - x,  multiplied 
by  the  coefficients  f( 1)  and/(0)  respectively.  And  the  expan- 
sion of  f(x,y)  consists  of  the  four  terms  xy,  x (1  -y),  (1  - x)  y, 
and  (1  - x),  (1  - y),  multiplied  by  the  coefficients  /( 1, 1),  /(l,  0), 
f(0,  1),  /( 0,  0),  respectively.  The  terms  x,  1 - x,  in  the  former 
case,  and  the  terms  xy,  a;(l  - y),  &c.,  in  the  latter,  we  shall  call 
the  constituents  of  the  expansion.  It  is  evident  that  they  are  in 
form  independent  of  the  form  of  the  function  to  be  expanded . 
Of  the  constituent  xy , x and  y are  termed  the  factors . 

The  general  rule  of  development  will  therefore  consist  of  two 
parts,  the  first  of  which  will  relate  to  the  formation  of  the  consti- 
tuents of  the  expansion,  the  second  to  the  determination  of  their 
respective  coefficients.  It  is  as  follows : 

1st.  To  expand  any  function  of  the  symbols  x,  y,  z. — Form  a 
series  of  constituents  in  the  following  manner  : Let  the  first  con- 
stituent be  the  product  of  the  symbols ; change  in  this  product 
any  symbol  z into  1 - z,  for  the  second  constituent.  Then  in 
both  these  change  any  other  symbol  y into  1 - y,  for  two  more 
constituents.  Then  in  the  four  constituents  thus  obtained  change 
any  other  symbol  x into  1 - x,  for  four  new  constituents,  and  so 
on  until  the  number  of  possible  changes  is  exhausted. 

2ndly.  To  find  the  coefficient  of  any  constituent. — If  that  con- 


76  PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 

stituent  involves  x as  a factor,  change  in  the  original  function  x 
into  1 ; but  if  it  involves  1 - x as  a factor,  change  in  the  original 
function  x into  0.  Apply  the  same  rule  with  reference  to  the 
symbols  y,  z,  &c.  : the  final  calculated  value  of  the  function  thus 
transformed  will  be  the  coefficient  sought. 

The  sum  of  the  constituents,  multiplied  each  by  its  respective 
coefficient,  will  be  the  expansion  required. 

12.  It  is  worthy  of  observation,  that  a function  may  be  de- 
veloped with  reference  to  symbols  which  it  does  not  explicitly 
contain.  Thus  if,  proceeding  according  to  the  rule,  we  seek  to 
develop  the  function  1 - x,  with  reference  to  the  symbols  x and 
y,  we  have, 

When  x = 1 and  y = 1 the  given  function  = 0. 

® = 1.  „ V = 0 „ „ =0. 

x = 0 „ y = 1 „ „ =1. 

x = 0 „ y = 0 „ „ =1. 

Whence  the  development  is 

1 — x = 0 xy  + Q x (\  - y)  + ( \ - x)y  + (1-#)  (1  - y)  ; 
and  this  is  a true  development.  The  addition  of  the  terms  ( 1 - x)y 
and  (1  - x)  (1  - y)  produces  the  function  1 - x. 

The  symbol  1 thus  developed  according  to  the  rule,  with  re- 
spect to  the  symbol  x,  gives 

x + 1 — x. 

Developed  with  respect  to  x and  y,  it  gives 

xy  + .z(l  -y)  + (1  -x)  y + (l  - x)  (1  -y). 

Similarly  developed  with  respect  to  any  set  of  symbols,  it  pro- 
duces a series  consisting  of  all  possible  constituents  of  those 
symbols. 

13.  A few  additional  remarks  concerning  the  nature  of  the 
general  expansions  may  with  propriety  be  added.  Let  us  take, 
for  illustration,  the  general  theorem  (5),  which  presents  the  type 
of  development  for  functions  of  two  logical  symbols. 

In  the  first  place,  that  theorem  is  perfectly  true  and  intel- 
ligible when  x and  y are  quantitative  symbols  of  the  species  con- 
sidered in  this  chapter,  whatever  algebraic  form  may  be  assigned 
to  the  function  f(x,  y),  and  it  may  therefore  be  intelligibly  cm- 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING.  77 

ployed  in  any  stage  of  the  process  of  analysis  intermediate  be- 
tween the  change  of  interpretation  of  the  symbols  from  the 
logical  to  the  quantitative  system  above  referred  to,  and  the  final 
restoration  of  the  logical  interpretation. 

Secondly.  The  theorem  is  perfectly  true  and  intelligible  when 
x and  y are  logical  symbols,  provided  that  the  form  of  the  func- 
tion/ (x,  y ) is  such  as  to  represent  a class  or  collection  of  things, 
in  which  case  the  second  member  is  always  logically  interpretable. 
For  instance,  if f(x,  y)  represent  the  function  1 - x + xy , we  ob- 
tain on  applying  the  theorem 

1 - x + xy  = xy  + Q x (\  -?/)  + (!-  x)  y + (\  - x)  (\  - y), 

= xy  + (1  -x)y  + (1  -x)  (1  -y), 
and  this  result  is  intelligible  and  true. 

Thus  we  may  regard  the  theorem  as  true  and  intelligible  for 
quantitative  symbols  of  the  species  above  described,  alivays  ; for 
logical  symbols,  always  when  interpretable . Whensoever  there- 
fore it  is  employed  in  this  work  it  must  be  understood  that  the 
symbols  x,  y are  quantitative  and  of  the  particular  species  referred 
to,  if  the  expansion  obtained  is  not  interpretable. 

But  though  the  expansion  is  not  always  immediately  inter- 
pretable, it  always  conducts  us  at  once  to  results  which  are  in- 
terpretable. Thus  the  expression  x - y gives  on  development 
the  form 

which  is  not  generally  interpretable.  We  cannot  take,  in  thought, 
from  the  class  of  things  which  are  x's  and  not  y’s,  the  class  of 
things  which  are  y s and  not  x's,  because  the  latter  class  is  not 
contained  in  the  former.  But  if  the  form  x - y presented  itself 
as  the  first  member  of  an  equation,  of  which  the  second  member 
was  0,  we  should  have  on  development 

*(1  - V ) -3/C1  -*)  = °- 

Now  it  will  be  shown  in  the  next  chapter  that  the  above  equa- 
tion, x and  y being  regarded  as  quantitative  and  of  the  species 
described,  is  resolvable  at  once  into  the  two  equations 
x(\-y)  = 0,  y(  l-.r)  = 0, 

and  these  equations  are  directly  interpretable  in  Logic  when  lo- 


78 


PRINCIPLES  OF  SYMBOLICAL  REASONING.  [CHAP.  V. 

gical  interpretations  are  assigned  to  the  symbols  x and  y.  And 
it  may  be  remarked,  that  though  functions  do  not  necessarily  be- 
come interpretable  upon  development,  yet  equations  are  always 
reducible  by  this  process  to  interpretable  forms. 

14.  The  following  Proposition  establishes  some  important 
properties  of  constituents.  In  its  enunciation  the  symbol  t is 
employed  to  represent  indifferently  any  constituent  of  an  expan- 
sion. Thus  if  the  expansion  is  that  of  a function  of  two  symbols 
x and  y,  t represents  any  of  the  four  forms  xy,  x (1  - y),  (1  - x)y, 
and  (l  - x)  (1  - y).  Where  it  is  necessary  to  represent  the  con- 
stituents of  an  expansion  by  single  symbols,  and  yet  to  distinguish 
them  from  each  other,  the  distinction  will  be  marked  by  suffixes. 
Thus  ti  might  be  employed  to  represent  xy,  t2  to  represent  x ( 1 - y), 
and  so  on. 

Proposition  III. 

Any  single  constituent  t of  an  expansion  satisfies  the  law  of  dua- 
lity whose  expression  is 

t(l-t)  = 0. 

The  product  of  any  two  distinct  constituents  of  an  expansion  is  equal 
to  0,  and  the  sum  of  all  the  constituents  is  equal  to  1. 

1st.  Consider  the  particular  constituent  xy.  We  have 

xy  x xy  = x2  y2. 

But  x2  = x,  y2  = y,  by  the  fundamental  law  of  class  symbols ; 
hence 

xy  x xy  = xy. 

Or  representing  xy  by  t, 

t x ij  = ij] 

or  if  (1  - t ) = 0. 

Similarly  the  constituent  x ( 1 - y)  satisfies  the  same  law.  For  we 
have 

x2  = x,  (1  - y)2  = 1 - y, 

.-.  {*(1  - y)}2  = z(l  -y),  or  £(1  -t)  = 0. 

Now  every  factor  of  every  constituent  is  either  of  the  form  x or 
of  the  form  1 - x.  Hence  the  square  of  each  factor  is  equal  to  that 


CHAP.  V.]  PRINCIPLES  OF  SYMBOLICAL  REASONING. 


79 


factor,  and  therefore  the  square  of  the  product  of  the  factors,  i.  e. 
of  the  constituent,  is  equal  to  the  constituent ; wherefore  t repre- 
senting any  constituent,  we  have 

tr  = t,  or  t (1  - t)  = 0. 

2ndly.  The  product  of  any  two  constituents  is  0.  This  is 
evident  from  the  general  law  of  the  symbols  expressed  by  the 
equation  x (1  - x)  = 0 ; for  whatever  constituents  in  the  same  ex- 
pansion we  take,  there  will  be  at  least  one  factor  x in  the  one,  to 
which  will  correspond  a factor  1 - x in  the  other. 

3rdly.  The  sum  of  all  the  constituents  of  an  expansion  is 
unity.  This  is  evident  from  addition  of  the  two  constituents  x 
and  1 - x,  or  of  the  four  constituents,  xy,  x (1  - y),  (1  - x)  y, 
(1  - x)  (1  - y).  But  it  is  also,  and  more  generally,  proved  by 
expanding  1 in  terms  of  any  set  of  symbols  (V.  12).  The  consti- 
tuents in  this  case  are  formed  as  usual,  and  all  the  coefficients 
are  unity. 

15.  With  the  above  Proposition  we  may  connect  the  fol- 
lowing. 

Proposition  IV. 

If  V represent  the  sum  of  any  series  of  constituents , the  separate 
coefficients  of  which  are  1,  then  is  the  condition  satisfied, 

V(l-V)  = 0. 

Let  t\,  . . . tn  be  the  constituents  in  question,  then 

V — t\  + t2  . . . 4-  tn. 

Squaring  both  sides,  and  observing  that  t?  - tu  tx  t2  = 0,  &c.,  we 
have 

V2  = tx  + t2  . . . + tn  ; 

whence 


Therefore 


V = F2. 

F(1  - V)  = 0. 


80 


OF  INTERPRETATION. 


[CHAP.  VI. 


CHAPTEK  VI. 

OF  THE  GENERAL  INTERPRETATION  OF  LOGICAL  EQUATIONS,  AND 
THE  RESULTING  ANALYSIS  OF  PROPOSITIONS.  ALSO,  OF  THE 
CONDITION  OF  INTERPRET  ABILITY  OF  LOGICAL  FUNCTIONS. 

1.  TT  has  been  observed  that  the  complete  expansion  of  any 
function  by  the  general  rule  demonstrated  in  the  last 
chapter,  involves  two  distinct  sets  of  elements,  viz.,  the  consti- 
tuents of  the  expansion,  and  their  coefficients.  I propose  in 
the  present  chapter  to  inquire,  first,  into  the  interpretation  of 
constituents,  and  afterwards  into  the  mode  in  which  that  inter- 
pretation is  modified  by  the  coefficients  with  which  they  are 
connected. 

The  terms  “ logical  equation,”  “ logical  function,”  &c.,  will 
be  . employed  generally  to  denote  any  equation  or  function  in- 
volving the  symbols  x , y,  &c.,  which  may  present  itself  either 
in  the  expression  of  a system  of  premises,  or  in  the  train  of  sym- 
bolical results  which  intervenes  between  the  premises  and  the 
conclusion.  If  that  function  or  equation  is  in  a form  not  imme- 
diately interpretable  in  Logic,  the  symbols  x,  y,  &c.,  must  be  re- 
garded as  quantitative  symbols  of  the  species  described  in  previous 
chapters  (II.  15),  (V.  6),  as  satisfying  the  law, 

x (1  - x)  = 0. 

By  the  problem,  then,  of  the  interpretation  of  any  such  logical 
function  or  equation,  is  meant  the  reduction  of  it  to  a form  in 
which,  when  logical  values  are  assigned  to  the  symbols  x,  y,  &c., 
it  shall  become  interpretable,  together  with  the  resulting  inter- 
pretation. These  conventional  definitions  are  in  accordance  with 
the  general  principles  for  the  conducting  of  the  method  of  this 
treatise,  laid  down  in  the  previous  chapter. 


CHAP.  VI.] 


OF  INTERPRETATION. 


81 


Proposition  I. 

2.  The  constituents  of  the  expansion  of  any  function  of  the  logi- 
cal symbols  x,  y,  8fc.,  are  interpretable , and  represent  the  several 
exclusive  divisions  of  the  universe  of  discourse , formed  by  the  predica- 
tion and  denial  in  every  possible  way  of  the  qualities  denoted  by  the 
symbols  x,  y,  fc. 

For  greater  distinctness  of  conception,  let  it  be  supposed  that 
the  function  expanded  involves  two  symbols  x and  y,  with  re- 
ference to  which  the  expansion  has  been  effected.  W e have  then 
the  following  constituents,  viz. : 

xy,  x(l  -y),  (l-x)y,  (l-x)(l-y). 

Of  these  it  is  evident,  that  the  first  xy  represents  that  class 
of  objects  which  at  the  same  time  possess  both  the  elementary 
qualities  expressed  by  x and  y,  and  that  the  second  x (l  - y)  re- 
presents the  class  possessing  the  property  x,  but  not  the  property 
y.  In  like  manner  the  third  constituent  represents  the  class  of 
objects  which  possess  the  property  represented  by  y,  but  not 
that  represented  by  x ; and  the  fourth  constituent  (1  - x)  (1  - y), 
represents  that  class  of  objects,  the  members  of  which  possess  nei- 
ther of  the  qualities  in  question. 

Thus  the  constituents  in  the  case  just  considered  represent 
all  the  four  classes  of  objects  which  can  be  described  by  affirma- 
tion and  denial  of  the  properties  expressed  by  x and  y.  Those 
classes  are  distinct  from  each  other.  No  member  of  one  is  a mem- 
ber of  another,  for  each  class  possesses  some  property  or  quality 
contrary  to  a property  or  quality  possessed  by  any  other  class. 
Again,  these  classes  together  make  up  the  universe,  for  there  is 
no  object  which  may  not  be  described  by  the  presence  or  the 
absence  of  a proposed  quality,  and  thus  each  individual  thing  in 
the  universe  may  be  referred  to  some  one  or  other  of  the  four 
classes  made  by  the  possible  combination  of  the  two  given 
classes  x and  y,  and  their  contraries. 

The  remarks  which  have  here  been  made  with  reference  to  the 
constituents  of  / ( x , y)  are  perfectly  general  in  character.  The 
constituents  of  any  expansion  represent  classes — those  classes 


82  OF  INTERPRETATION.  [CHAP.  VI. 

are  mutually  distinct,  through  the  possession  of  contrary  qualities, 
and  they  together  make  up  the  universe  of  discourse. 

3.  These  properties  of  constituents  have  their  expression  in 
the  theorems  demonstrated  in  the  conclusion  of  the  last  chapter, 
and  might  thence  have  been  deduced.  From  the  fact  that  every 
constituent  satisfies  the  fundamental  law  of  the  individual  sym- 
bols, it  might  have  been  conjectured  that  each  constituent  would 
represent  a class.  From  the  fact  that  the  product  of  any  two 
constituents  of  an  expansion  vanishes,  it  might  have  been  con- 
cluded that  the  classes  they  represent  are  mutually  exclusive. 
Lastly,  from  the  fact  that  the  sum  of  the  constituents  of  an  ex- 
pansion is  unity,  it  might  have  been  inferred,  that  the  classes 
which  they  represent,  together  make  up  the  universe. 

4.  Upon  the  laws  of  constituents  and  the  mode  of  their  in- 
terpretation above  determined,  are  founded  the  analysis  and  the 
interpretation  of  logical  equations.  That  all  such  equations  ad- 
mit of  interpretation  by  the  theorem  of  development  has  already 
been  stated.  I propose  here  to  investigate  the  forms  of  possible 
solution  which  thus  present  themselves  in  the  conclusion  of  a 
train  of  reasoning,  and  to  show  how  those  forms  arise.  Although, 
properly  speaking,  they  are  but  manifestations  of  a single  funda- 
mental type  or  principle  of  expression,  it  will  conduce  to  clearness 
of  apprehension  if  the  minor  varieties  which  they  exhibit  are 
presented  separately  to  the  mind. 

The  forms,  which  are  three  in  number,  are  as  follows  : 

FORM  i. 

5.  The  form  we  shall  first  consider  arises  when  any  logical 
equation  V=  0 is  developed,  and  the  result,  after  resolution  into 
its  component  equations,  is  to  be  interpreted.  The  function  is  sup- 
posed to  involve  the  logical  symbols  x,y,&c.,  in  combinations  which 
are  not  fractional.  Fractional  combinations  indeed  only  arise  in 
the  class  of  problems  which  will  be  considered  when  we  come  to 
speak  of  the  third  of  the  forms  of  solution  above  referred  to. 

Proposition  II. 

To  interpret  the  logical  equation  V=  0. 

For  simplicity  let  us  suppose  that  V involves  but  two  sym- 


OF  INTERPRETATION. 


83 


CHAP.  VI.] 

bols,  x and  y,  and  let  us  represent  the  development  of  the  given 
equation  by 

axy  + bx  (1  - y)  + c (1  - x)  y + <2(1  - x)  (1  - y)  = 0;  (1) 

a,  b,  c,  and  d being  definite  numerical  constants. 

Now,  suppose  that  any  coefficient,  as  a,  does  not  vanish. 
Then  multiplying  each  side  of  the  equation  by  the  constituent  xy , 
to  which  that  coefficient  is  attached,  we  have 

axy  = 0, 

whence,  as  a does  not  vanish, 

xy  = 0, 

and  this  result  is  quite  independent  of  the  nature  of  the  other  co- 
efficients of  the  expansion.  Its  interpretation,  on  assigning  to 
x and  y their  logical  significance,  is  “No  individuals  belonging  at 
once  to  the  class  represented  by  x,  and  the  class  represented  by  y, 
exist.” 

But  if  the  coefficient  a does  vanish,  the  term  axy  does  not 
appear  in  the  development  (1),  and,  therefore,  the  equation  xy  = 0 
cannot  thence  be  deduced. 

In  like  manner,  if  the  coefficient  b does  not  vanish,  we  have 

* (i  - y)  = o, 

which  admits  of  the  interpretation,  “ There  are  no  individuals 
which  at  the  same  time  belong  to  the  class  x,  and  do  not  belong 
to  the  class  y.” 

Either  of  the  above  interpretations  may,  however,  as  will  sub- 
sequently be  shown,  be  exhibited  in  a different  form. 

The  sum  of  the  distinct  interpretations  thus  obtained  from 
the  several  terms  of  the  expansion  whose  coefficients  do  not 
vanish,  will  constitute  the  complete  interpretation  of  the  equation 
V = 0.  The  analysis  is  essentially  independent  of  the  number 
of  logical  symbols  involved  in  the  function  V,  and  the  object  of 
the  proposition  will,  therefore,  in  all  instances,  be  attained  by  the 
following  Rule: — 

Rule. — Develop  the  function  V,  and  equate  to  0 every  consti- 
tuent whose  coefficient  does  not  vanish.  The  interpretation  of  these 
results  collectively  will  constitute  the  interpretation  of  the  yiven 
equation . 


84 


OF  INTERPRETATION. 


[CHAP.  VI. 

6.  Let  us  take  as  an  example  the  definition  of  “ clean  beasts,” 
laid  down  in  the  Jewish  law,  viz.,  “ Clean  beasts  are  those 
which  both  divide  the  hoof  and  chew  the  cud,”  and  let  us  assume 

x = clean  beasts  ; 
y = beasts  dividing  the  hoof; 
z = beasts  chewing  the  cud. 

Then  the  given  proposition  will  be  represented  by  the  equation 

x = yz, 

which  we  shall  reduce  to  the  form 

x - yz  = 0, 

and  seek  that  form  of  interpretation  to  which  the  present  method 
leads.  Fully  developing  the  first  member,  we  have 

0 xyz  + xy  ( 1 — z)  + x ( 1 — y)  z + x ( 1 — y)  ( 1 - z) 
-(1-^)y2+°(l-^)y(l-^)  + 0(1-.r)(l-y)2  + °(1-^)(1-y)(l-2). 

Whence  the  terms,  whose  coefficients  do  not  vanish,  give 

zy(\-z)  = 0,  xz(\-y)  = 0,  x(\-y)(\-z)  = 0,  (\-x)yz  = 0. 

These  equations  express  a denial  of  t he  existence  of  certain  classes 
of  objects,  viz. : 

1st.  Of  beasts  which  are  clean,  and  divide  the  hoof,  but  do 
not  chew  the  cud. 

2nd.  Of  beasts  which  are  clean,  and  chew  the  cud,  but  do  not 
divide  the  hoof. 

3rd.  Of  beasts  which  are  clean,  and  neither  divide  the  hoof 
nor  chew  the  cud. 

4th.  Of  beasts  which  divide  the  hoof,  and  chew  the  cud,  and 
are  not  clean. 

Now  all  these  several  denials  are  really  involved  in  the  origi- 
nal proposition.  And  conversely,  if  these  denials  be  granted, 
the  original  proposition  will  follow  as  a necessary  consequence. 
They  are,  in  fact,  the  separate  elements  of  that  proposition. 
Every  primary  proposition  can  thus  be  resolved  into  a series  of 
denials  of  the  existence  of  certain  defined  classes  of  things,  and 
may,  from  that  system  of  denials,  be  itself  reconstructed.  It 
might  here  be  asked,  how  it  is  possible  to  make  an  assertive  pro- 


CHAP.  VI.] 


OF  INTERPRETATION. 


85 


position  out  of  a series  of  denials  or  negations  ? From  what 
source  is  the  positive  element  derived  ? I answer,  that  the  mind 
assumes  the  existence  of  a universe  not  a priori  as  a fact  inde- 
pendent of  experience,  but  either  a,  posteriori  as  a deduction 
from  experience,  or  hypothetically  as  a foundation  of  the  possi- 
bility of  assertive  reasoning.  Thus  from  the  Proposition,  “There 
are  no  men  who  are  not  fallible,”  which  is  a negation  or  denial  of 
the  existence  of  “ infallible  men,”  it  may  be  inferred  either  hypo- 
thetically, “ All  men  (if  men  exist)  are  fallible,”  or  absolutely, 
(experience  having  assured  us  of  the  existence  of  the  race),  “ All 
men  are  fallible.” 

The  form  in  Avhich  conclusions  are  exhibited  by  the  method 
of  this  Proposition  may  be  termed  the  form  of  “ Single  or  Con- 
joint Denial.” 

FORM  II. 

7.  As  the  previous  form  was  derived  from  the  development 
and  interpretation  of  an  equation  whose  second  member  is  0,  the 
present  form,  which  is  supplementary  to  it,  will  be  derived  from 
the  development  and  interpretation  of  an  equation  whose  second 
member  is  1.  It  is,  however,  readily  suggested  by  the  analysis 
of  the  previous  Proposition. 

Thus  in  the  example  last  discussed  we  deduced  from  the 
equation 

x - yz  = 0 

the  conjoint  denial  of  the  existence  of  the  classes  represented  by 
the  constituents 

xy{\-z),  xz{\  - y),  x (1  - y)  (1  - z),  (l-x)yz, 

whose  coefficients  were  not  equal  to  0.  It  follows  hence  that 
the  remaining  constituents  represent  classes  which  make  up  the 
universe.  Hence  we  shall  have 

*y*  + (i  -x)y{}  -*)  + 0 -*)  (l -y)z  + (l  -•*)  (i  -y)  (i  ~z)  = 1. 

This  is  equivalent  to  the  affirmation  that  all  existing  things  be- 
long to  some  one  or  other  of  the  following  classes,  viz. : 

1st.  Clean  beasts  both  dividing  the  hoof  and  chewing  the 
cud. 


86  OF  INTERPRETATION.  [CHAP.  VI. 

2nd.  Unclean  beasts  dividing  the  hoof,  but  not  chewing  the 
cud. 

3rd.  Unclean  beasts  chewing  the  cud,  but  not  dividing  the 
hoof. 

4th.  Things  which  are  neither  clean  beasts,  nor  chewers  of 
the  cud,  nor  dividers  of  the  hoof. 

This  form  of  conclusion  may  be  termed  the  form  of  “ Single 
or  Disjunctive  Affirmation,” — single  when  but  one  constituent 
appears  in  the  final  equation;  disj  unctive  when,  as  above,  more 
constituents  than  one  are  there  found. 

Any  equation,  V = 0,  wherein  V satisfies  the  law  of  duality, 
may  also  be  made  to  yield  this  form  of  interpretation  by  reducing 
it  to  the  form  1 - V=  1,  and  developing  the  first  member.  The 
case,  however,  is  really  included  in  the  next  general  form.  Both 
the  previous  forms  are  of  slight  importance  compared  with  the 
following  one. 

FORM  III. 

8.  In  the  two  preceding  cases  the  functions  to  be  developed 
were  equated  to  0 and  to  1 respectively.  In  the  present  case  I 
shall  suppose  the  corresponding  function  equated  to  any  logical 
symbol  w.  We  are  then  to  endeavour  to  interpret  the  equation 
V = w,  V being  a function  of  the  logical  symbols  x,  y,  z,  &c.  In 
the  first  place,  however,  I deem  it  necessary  to  show  how  the 
equation  V = w,  or,  as  it  will  usually  present  itself,  w = V,  arises. 

Let  us  resume  the  definition  of  “ clean  beasts,”  employed  in 
the  previous  examples,  viz.,  “Clean  beasts  are  those  which  both 
divide  the  hoof  and  chew  the  cud,”  and  suppose  it  required  to  de- 
termine the  relation  in  which  “ beasts  chewing  the  cud”  stand  to 
“ clean  beasts”  and  “beasts  dividing  the  hoof.”  The  equation 
expressing  the  given  proposition  is 

x = yz , 

and  our  object  will  be  accomplished  if  we  can  determine  z as  an 
interpretable  function  of  x and  y. 

Now  treating  x,  y,  z as  symbols  of  quantity  subject  to  a pe- 
culiar law,  we  may  deduce  from  the  above  equation,  by  solution, 

x 

' ~ y 


CHAP.  VI.] 


OF  INTERPRETATION. 


87 


But  this  equation  is  not  at  present  in  an  interpretable  form.  If 
we  can  reduce  it  to  such  a form  it  will  furnish  the  relation 
required. 

On  developing  the  second  member  of  the  above  equation,  we 
have 

* = *y  + q * (1  - V)  + 0 (1  - x)  y + jj  (1  - x)  (1  - y), 


and  it  will  be  shown  hereafter  (Prop.  3)  that  this  admits  of  the 
following  interpretation : 

“ Beasts  which  chew  the  cud  consist  of  all  clean  beasts 
(which  also  divide  the  hoof),  together  with  an  indefinite  re- 
mainder (some,  none,  or  all)  of  unclean  beasts  which  do  not  di- 
vide the  hoof.” 

9.  Now  the  above  is  a particular  example  of  a problem  of  the 
utmost  generality  in  Logic,  and  which  may  thus  be  stated : — 
“ Given  any  logical  equation  connecting  the  symbols  x,  y,  z,  tv, 
required  an  interpretable  expression  for  the  relation  of  the  class 
represented  by  to  to  the  classes  represented  by  the  other  symbols 
x,  y,  z,  &c.” 

The  solution  of  this  problem  consists  in  all  cases  in  deter- 
mining, from  the  equation  given , the  expression  of  the  above 
symbol  to , in  terms  of  the  other  symbols,  and  rendering  that  ex- 
pression interpre table  by  development.  Now  the  equation  given 
is  always  of  the  first  degree  with  respect  to  each  of  the  symbols 
involved.  The  required  expression  for  to  can  therefore  always 
be  found.  In  fact,  if  we  develop  the  given  equation,  whatever 
its  form  may  be  with  respect  to  tv,  we  obtain  an  equation  of  the 
form 

Etv  + E'  (1  - to)  = 0,  (1) 


E and  E’  being  functions  of  the  remaining  symbols, 
above  we  have 

E = (E  - E)  to. 

Therefore 


w 


E 

E - E 


From  the 


(2) 


and  expanding  the  second  member  by  the  rule  of  development,  it 
will  only  remain  to  interpret  the  result  in  logic  by  the  next 
proposition. 


88 


OF  INTERPRETATION. 


[CHAP.  VI. 


If  the  fraction 


E 


has  common  factors  in  its  numerator 


E - E 

and  denominator,  we  are  not  pei'mitted  to  reject  them,  unless  they 
are  mere  numerical  constants.  For  the  symbols  x , y,  &c.,  re- 
garded as  quantitative,  may  admit  of  such  values  0 and  1 as  to 
cause  the  common  factors  to  become  equal  to  0,  in  which  case 
the  algebraic  rule  of  reduction  fails.  This  is  the  case  contem- 
plated in  our  remarks  on  the  failure  of  the  algebraic  axiom  of 
division  (II.  14).  To  express  the  solution  in  the  form  (2),  and 
without  attempting  to  perform  any  unauthorized  reductions,  to 
interpret  the  result  by  the  theorem  of  development,  is  a course 
strictly  in  accordance  with  the  general  principles  of  this  treatise. 

If  the  relation  of  the  class  expressed  by  1 - xo  to  the  other 
classes,  x,  y,  &c.  is  required,  we  deduce  from  (1),  in  like  manner 
as  above, 

E 


1 - w = 


E - E 


to  the  interpretation  of  which  also  the  method  of  the  following 
Proposition  is  applicable  : 


Proposition  III. 

10.  To  determine  the  interpretation  of  any  logical  equation  of 
the  form  w = V,  in  wh  ich  w is  a class  symbol,  and  V a function  of 
other  class  symbols  quite  unlimited  in  its  form. 

Let  the  second  member  of  the  above  equation  be  fully  ex- 
panded. Each  coefficient  of  the  result  will  belong  to  some  one 
of  the  four  classes,  which,  with  their  respective  interpretations, 
we  proceed  to  discuss. 

1st.  Let  the  coefficient  be  1.  As  this  is  the  symbol  of  the 
universe,  and  as  the  product  of  any  two  class  symbols  represents 
those  individuals  which  are  found  in  both  classes,  any  constituent 
which  has  unity  for  its  coefficient  must  be  interpreted  without 
limitation,  i.  e.  the  whole  of  the  class  which  it  represents  is 
implied. 

2nd.  Let  the  coefficient  be  0.  As  in  Logic,  equally  with 
Arithmetic,  this  is  the  symbol  of  Nothing,  no  part  of  the  class 


OF  INTERPRETATION. 


89 


CHAP.  VI.] 

represented  by  the  constituent  to  which  it  is  prefixed  must  be 
taken. 

3rd.  Let  the  coefficient  be  of  the  form  jj.  Now,  as  in  Arith- 
metic, the  symbol  ^ represents  an  indefinite  number , except  when 

otherwise  determined  by  some  special  circumstance,  analogy 
would  suggest  that  in  the  system  of  this  work  the  same  symbol 
should  represent  an  indefinite  class.  That  this  is  its  true  mean- 
ing will  be  made  clear  from  the  following  example  : 

Let  us  take  the  Proposition,  “ Men  not  mortal  do  not  exist 
represent  this  Proposition  by  symbols ; and  seek,  in  obedience  to 
the  laws  to  which  those  symbols  have  been  proved  to  be  subject, 
a reverse  definition  of  “ mortal  beings,”  in  terms  of  “ men.” 

Now  if  we  represent  “ men”  by  y,  and  “ mortal  beings”  by  x, 
the  Proposition,  “Men  who  are  not  mortals  do  not  exist,”  will 
be  expressed  by  the  equation 

y(l  - x)  = 0, 

from  which  we  are  to  seek  the  value  of  x.  Now  the  above  equa- 
tion gives 

y-yx  = 0,  or  yx  = y. 

Were  this  an  ordinary  algebraic  equation,  we  should,  in  the  next 
place,  divide  both  sides  of  it  by  y.  But  it  has  been  remarked  in 
Chap.  ii.  that  the  operation  of  division  cannot  be  performed  with 
the  symbols  with  which  we  are  now  engaged.  Our  resource,  then, 
is  to  express  the  operation,  and  develop  the  result  by  the  method 
of  the  preceding  chapter.  We  have,  then,  first, 


and,  expanding  the  second  member  as  directed, 

* = V + 5 (l  “ y)‘ 

This  implies  that  mortals  (x)  consist  of  all  men  (y),  together 
with  such  a remainder  of  beings  which  are  not  men  (l  - y),  as 

Avill  be  indicated  by  the  coefficient  Now  let  us  inquire  what 


90 


OF  INTERPRETATION. 


[CHAP.  VI. 


remainder  of  “ not  men”  is  implied  by  the  premiss.  It  might 
happen  that  the  remainder  included  all  the  beings  who  are  not 
men,  or  it  might  include  only  some  of  them,  and  not  others,  or  it 
might  include  none,  and  any  one  of  these  assumptions  would  be 
in  perfect  accordance  with  our  premiss.  In  other  words,  whether 
those  beings  which  are  not  men  are  all,  or  some,  or  none,  of  them 
mortal,  the  truth  of  the  premiss  which  virtually  asserts  that  all 
men  are  mortal,  will  be  equally  unaffected,  and  therefore  the 


expression  - here  indicates  that  all,  some , or  none  of  the  class  to 

whose  expression  it  is  affixed  must  be  taken. 

Although  the  above  determination  of  the  significance  of  the 

symbol  - is  founded  only  upon  the  examination  of  a particular 


case,  yet  the  principle  involved  in  the  demonstration  is  general, 
and  there  are  no  circumstances  under  which  the  symbol  can  pre- 
sent itself  to  which  the  same  mode  of  analysis  is  inapplicable. 

We  may  properly  term  - an  indefinite  class  symbol,  and  may,  if 


convenience  should  require,  replace  it  by  an  uncompounded  sym- 
bol v,  subject  to  the  fundamental  law,  v (1  - v)  = 0. 

4th.  It  may  happen  that  the  coefficient  of  a constituent  in  an 
expansion  does  not  belong  to  any  of  the  previous  cases.  To  as- 
certain its  true  interpretation  when  this  happens,  it  will  be  ne- 
cessary to  premise  the  following  theorem : 

11.  Theorem. — If  a function  V,  intended  to  represent  any 
class  or  collection  of  objects,  ic,  be  expanded,  and  if  the  numerical 
coefficient,  a,  of  any  constituent  in  its  development , do  not  satisfy 
the  law. 

a (1  - a)  = 0, 


then  the  constituent  in  question  must  be  made  equal  to  0. 

To  prove  the  theorem  generally,  let  us  represent  the  expan- 
sion given,  under  the  form 

w = ajtj  + a2t2  + a3t3  + &c.,  (1) 


in  which  tu  t2,  t3,  &c.  represent  the  constituents,  and  au  a2,  a3,  &c. 
the  coefficients ; let  us  also  suppose  that  a x and  a2  do  not  satisfy 
the  law 


U\  [1  — ajj  — 0,  a2  [1  aj)  - 0 , 


OF  INTERPRETATION. 


91 


CHAP.  VI.] 

but  that  the  other  coefficients  are  subject  to  the  law  in  question, 
so  that  we  have 

«32  = a3,  &c. 

Now  multiply  each  side  of  the  equation  (1)  by  itself.  The  re- 
sult will  be 

w = a !2  tx  + a22 t2  + &c.  (2) 

This  is  evident  from  the  fact  that  it  must  represent  the  develop- 
ment of  the  equation 

w = V2, 

but  it  may  also  be  proved  by  actually  squaring  (1),  and  observing 
that  we  have 

t2  = tu  t2 2 = t2,  tx  t2  = 0,  &c. 

by  the  properties  of  constituents.  Now  subtracting  (2)  from  (1), 
we  have 

(ax  - ax2)  tx  + (<z2  - a22)  t2  = 0. 

Or,  ax  (1  - ax)  tx  + a2  (1  - a2)  t*.  = 0. 

Multiply  the  last  equation  by  U ; then  since  tx  t2  - 0,  we  have 
ax  (1  - ax)  tx  = 0,  whence  tx  = 0. 

In  like  manner  multiplying  the  same  equation  by  (2,  we  have 
a2  ( 1 - a2)  t2  = 0,  whence  t2  = 0. 

Thus  it  may  be  shown  generally  that  any  constituent  whose 
coefficient  is  not  subject  to  the  same  fundamental  law  as  the  sym- 
bols themselves  must  be  separately  equated  to  0.  The  usual 

form  under  which  such  coefficients  occur  is  This  is  the  alge- 
braic symbol  of  infinity.  Now  the  nearer  any  number  approaches 
to  infinity  (allowing  such  an  expression),  the  more  does  it  depart 
from  the  condition  of  satisfying  the  fundamental  law  above  re- 
ferred to. 

0 

The  symbol  -,  whose  interpretation  was  previously  dis- 
cussed, does  not  necessarily  disobey  the  law  we  are  here  consi- 
dering, for  it  admits  of  the  numerical  values  0 and  1 indifferently. 
Its  actual  interpretation,  however,  as  an  indefinite  class  symbol, 
cannot,  I conceive,  except  upon  the  ground  of  analogy,  be  de- 


OF  INTERPRETATION. 


92 


[chap.  VI. 


duced  from  its  arithmetical  properties,  but  must  be  established 
experimentally. 

12.  We  may  now  collect  the  results  to  which  we  have  been 
led,  into  the  following  summary : 

1st.  The  symbol  1,  as  the  coefficient  of  a term  in  a develop- 
ment, indicates  that  the  whole  of  the  class  which  that  constituent 
represents,  is  to  be  taken. 

2nd.  The  coefficient  0 indicates  that  none  of  the  class  are  to 
be  taken. 

3rd.  The  symbol  - indicates  that  a perfectly  indefinite  por- 
tion of  the  class,  i.  e.  some,  none,  or  all  of  its  members  are  to  be 
taken. 

4th.  Any  other  symbol  as  a coefficient  indicates  that  the 
constituent  to  which  it  is  prefixed  must  be  equated  to  0. 

It  follows  hence  that  if  the  solution  of  a problem,  obtained 
by  development,  be  of  the  form 


io  = A + OB  + H C + i D, 


that  solution  may  be  resolved  into  the  two  following  equations, 
viz., 

w = A + vC,  (3) 

D = 0,  (4) 

v being  an  indefinite  class  symbol.  The  interpretation  of  (3) 
shows  what  elements  enter,  or  may  enter,  into  the  composition 
of  w,  the  class  of  things  whose  definition  is  required ; and  the 
interpretation  of  (4)  shows  what  relations  exist  among  the  ele- 
ments of  the  original  problem,  in  perfect  independence  of  w. 

Such  are  the  canons  of  interpretation.  It  may  be  added,  that 
they  are  universal  in  their  application,  and  that  their  use  is 
always  unembarrassed  by  exception  or  failure. 

13.  Corollary. — If  Che  an  independently  interpretable  logi- 
cal function,  it  will  satisfy  the  symbolical  law,  V (l  - V)  = 0. 

By  an  independently  interpretable  logical  fimction,  I mean 
one  which  is  interpretable,  without  presupposing  any  relation 
among  the  things  represented  by  the  symbols  which  it  involves. 
Thus  x ( 1 - if)  is  independently  interpretable,  but  x - y is  not  so. 


OF  INTERPRETATION. 


93 


CHAP.  VI.] 

The  latter  function  presupposes,  as  a condition  of  its  interpreta- 
tion, that  the  class  represented  by  y is  wholly  contained  in  the 
class  represented  by  x ; the  former  function  does  not  imply  any 
such  requirement. 

Now  if  V be  independently  interpretable,  and  if  w represent 
the  collection  of  individuals  which  it  contains,  the  equation 
w = V will  hold  true  without  entailing  as  a consequence  the  va- 
nishing of  any  of  the  constituents  in  the  development  of  V ; 
since  such  vanishing  of  constituents  would  imply  relations  among 
the  classes  of  things  denoted  by  the  symbols  in  V.  Hence  the 
development  of  V will  be  of  the  form 

GL\  t\  + ^2  &C* 

the  coefficients  ax,  a2,  &c.  all  satisfying  the  condition 
ax  (1  - ax)  = 0,  g2  (1  - a2)  = 0,  &c. 

Hence  by  the  reasoning  of  Prop.  4,  Chap.  v.  the  function  V will 
be  subject  to  the  law 

F(1  - V)  = 0. 

This  result,  though  evident  a priori  from  the  fact  that  V is  sup- 
posed to  represent  a class  or  collection  of  things,  is  thus  seen  to 
follow  also  from  the  properties  of  the  constituents  of  which  it  is 
composed.  The  condition  F(1  - V)  = 0 may  be  termed  “the 
condition  of  interpretability  of  logical  functions.” 

14.  The  general  form  of  solutions,  or  logical  conclusions  de- 
veloped in  the  last  Proposition,  may  be  designated  as  a “ Relation 
between  terms.”  I use,  as  before,  the  word  “ terms”  to  denote 
the  parts  of  a proposition,  whether  simple  or  complex,  which  are 
connected  by  the  copula  “ is”  or  “ are.”  The  classes  of  things  re- 
presented by  the  individual  symbols  may  be  called  the  elements 
of  the  proposition. 

15.  Ex.  1. — Resuming  the  definition  of  “ clean  beasts,” 
(VI. 6),  required  a description  of  “unclean  beasts.” 

Here,  as  before,  x standing  for  “ clean  beasts,”  y for  “beasts 
dividing  the  hoof,”  r for  “ beasts  chewing  the  cud,”  we  have 

x^yz\  (5) 

whence 

1 - x = 1 - yz ; 

‘and  developing  the  second  member, 


94 


OF  INTERPRETATION. 


[chap.  VI. 

1 - x = y(l  - z)  + * (1-  y)  + (1  - y)  (1  - z); 

which  is  interpretable  into  the  following  Proposition:  Unclean 
beasts  are  all  which  divide  the  hoof  without  chewing  the  cud , all 
which  chew  the  cud  without  dividing  the  hoof  , and  all  which  neither 
divide  the  hoof  nor  chew  the  cud. 

Ex.  2. — The  same  definition  being  given,  required  a descrip- 
tion of  beasts  which  do  not  divide  the  hoof. 

From  the  equation  x - yz  we  have 


x 


therefore. 


1 -y  = 


and  developing  the  second  member, 

- 1 0 

1 - y = 0 xz  + — x (1  - z)  + (1  - x)  z + - (1  - x)  (1  - z). 

Here,  according  to  the  Rule,  the  term  whose  coefficients  is 
must  be  separately  equated  to  0,  whence  we  have 


1 - y = (1  - x)  z + v (1  - a?)  (1  - z), 
x (1  - z)  = 0 ; 

whereof  the  first  equation  gives  by  interpretation  the  Proposition : 
Beasts  which  do  not  divide  the  hoof  consist  of  all  unclean  beasts  which 
cheiv  the  cud , and  an  indefinite  remainder  (some,  none,  or  all ) of  un- 
clean beasts  ivhich  do  not  chew  the  cud. 

The  second  equation  gives  the  Proposition : There  are  no  clean 
beasts  which  do  not  chew  the  cud.  This  is  one  of  the  independent 
relations  above  referred  to.  We  sought  the  direct  relation  of 
“ Beasts  not  dividing  the  hoof,”  to  “ Clean  beasts  and  beasts 
which  chew  the  cud.”  It  happens,  however,  that  independently 
of  any  relation  to  beasts  not  dividing  the  hoof,  there  exists,  in 
virtue  of  the  premiss,  a separate  relation  between  clean  beasts 
and  beasts  which  chew  the  cud.  This  relation  is  also  necessarily 
given  by  the  process. 

Ex.  3. — Let  us  take  the  following  definition,  viz. : “ Respon- 
sible beings  are  all  rational  beings  who  are  either  free  to  act,  or 


OF  INTERPRETATION. 


95 


CHAP.  VI.] 

have  voluntarily  sacrificed  their  freedom,”  and  apply  to  it  the 
preceding  analysis. 

Let  x stand  for  responsible  beings. 
y ,,  rational  beings. 

z ,,  those  who  are  free  to  act, 

w ,,  those  who  have  voluntarily  sacrificed  their 

freedom  of  action. 

In  the  expression  of  this  definition  I shall  assume,  that  the 
two  alternatives  which  it  presents,  viz. : “ Rational  beings  free 
to  act,”  and  “ Rational  beings  whose  freedom  of  action  has  been 
voluntarily  sacrificed,”  are  mutually  exclusive,  so  that  no  indivi- 
duals are  found  at  once  in  both  these  divisions.  This  will  per- 
mit us  to  interpret  the  proposition  literally  into  the  language  of 
symbols,  as  follows : 

x = yz  + yiu.  (6) 

Let  us  first  determine  hence  the  relation  of ts  rational  beings”  to 
responsible  beings,  beings  free  to  act,  and  beings  whose  freedom 
of  action  has  been  voluntarily  abjured.  Perhaps  this  object  will 
be  better  stated  by  saying,  that  we  desire  to  express  the  relation 
among  the  elements  of  the  premiss  in  such  a form  as  will  enable 
us  to  determine  how  far  rationality  may  be  inferred  from  respon- 
sibility, freedom  of  action,  a voluntary  sacrifice  of  freedom,  and 
their  contraries. 

From  (6)  we  have 

x 

y z+  w 

and  developing  the  second  member,  but  rejecting  terms  whose 
coefficients  are  0, 

y = i xzw  + xz(\  - w)  + x(l  - z)  w + ^--x  (1  - z)  (1  - w) 

jL  v 

+ jj(l -*)(!-*)  (I-*). 

whence,  equating  to  0 the  terms  whose  coefficients  are  ^ and 
we  have 

y = xz  (1  - w)  + .r w (1  - z)  + v (1  - x)  (1  - z)  (1  - «0;  (7) 

xziv  = 0 ; (8) 


96 


OF  INTERPRETATION. 


[CHAP.  VI. 

(9) 


x (1  - z)  (1  - w)  = 0 ; 
whence  by  interpretation — 

Direct  Conclusion. — Rational  beings  are  all  responsible  beings 
who  are  either  free  to  act,  not  having  voluntarily  sacrificed  their  free- 
dom, or  not  free  to  act,  having  voluntarily  sacrificed  their  freedom, 
together  with  an  indefinite  remainder  (some,  none,  or  all)  of  beings 
not  responsible,  not  free , and  not  having  voluntarily  sacrificed  their 
freedom. 

First  Independent  Relation. — No  responsible  beings  are  at 
the  same  time  free  to  act,  and  in  the  condition  of  having  voluntarily 
sacrificed  their  freedom. 

Second. — No  responsible  beings  are  not  free  to  act,  and  at  the 
same  time  in  the  condition  of  not  having  sacrificed  their  freedom. 

The  independent  relations  above  determined  may,  however, 
be  put  in  another  and  more  convenient  form.  Thus  (8)  gives 

xw  = - = 0 z + ^ (1  - z),  on  development ; 
z 0 

or,  xw  = v (1  - z) ; (10) 

and  in  like  manner  (9)  gives 

/i  \ 0 0 n/1  x 

a?  (1  - w)  = YZ~z  = o 2 + 0 C1  - z) ; 
or,  x (1  -w)  = vz;  (11) 

and  (10)  and  (11)  interpreted  give  the  following  Propositions : 

1st.  Responsible  beings  who  have  voluntarily  sacrificed  their  free- 
dom are  not  free. 

2nd.  Responsible  beings  who  have  not  voluntarily  sacrificed  their 
freedom  are  free. 

These,  however,  are  merely  different  forms  of  the  relations 
before  determined. 

16.  In  examining,  these  results,  the  reader  must  bear  in  mind, 
that  the  sole  province  of  a method  of  inference  or  analysis,  is  to 
determine  those  relations  which  are  necessitated  by  the  connexion 
of  the  terms  in  the  original  proposition.  Accordingly,  in  esti- 
mating the  completeness  with  which  this  object  is  effected,  we 
have  nothing  whatever  to  do  with  those  other  relations  which 


CHAP.  VI.] 


0!F  INTERPRETATION. 


97 


may  be  suggested  to  our  minds  by  the  meaning  of  the  terms 
employed,  as  distinct  from  their  expressed  connexion.  Thus  it 
seems  obvious  to  remark,  that  “ They  who  have  voluntarily  sa- 
crificed their  freedom  are  not  free,”  this  being  a relation  implied 
in  the  very  meaning  of  the  terms.  And  hence  it  might  appear, 
that  the  first  of  the  two  independent  relations  assigned  by  the  me- 
thod is  on  the  one  hand  needlessly  limited,  and  on  the  other  hand 
superfluous.  However,  if  regard  be  had  merely  to  the  connexion 
of  the  terms  in  the  original  premiss,  it  will  be  seen  that  the  re- 
lation in  question  is  not  liable  to  either  of  these  charges.  The 
solution,  as  expressed  in  the  direct  conclusion  and  the  indepen- 
dent relations,  conjointly,  is  perfectly  complete,  without  being 
in  any  way  superfluous. 

If  we  wish  to  take  into  account  the  implicit  relation  above 
referred  to,  viz.,  “ They  who  have  voluntarily  sacrificed  their 
freedom  are  not  free,”  we  can  do  so  by  making  this  a distinct 
proposition,  the  proper  expression  of  which  would  be 

w = v (1  - z). 

This  equation  we  should  have  to  employ  together  with  that 
expressive  of  the  original  premiss.  The  mode  in  which  such  an 
examination  must  be  conducted  will  appear  when  we  enter  upon 
the  theory  of  systems  of  propositions  in  a future  chapter.  The 
sole  difference  of  result  to  which  the  analysis  leads  is,  that  the 
first  of  the  independent  relations  deduced  above  is  superseded. 

17*  Ex.  4. — Assuming  the  same  definition  as  in  Example  2, 
let  it  be  required  to  obtain  a description  of  irrational  persons. 

We  have 


Z + W - X 
Z + 10 

= - xziv  + 0 xz  (1  - w)  + 0 X (1  - z)  W - ^ X (1  - z)  (1  - w) 

& V 

+ (l-*)2ia  + (l-a;).c(l-ir)  + (l-a;)(l-r)ty+jj(l-a:)(l-2)(l-M;) 

= (1-  x) 2i04(l-a:)2(l- w)  + (l-  .r)(l-  2)i»+v(l-;r)(l-2)(l-M>) 
= (l-a:)£+(l-;r)(l-2)u;  + i;(l-a:)(l-z)(l  - iv), 

with  xzw  = 0,  ;r(l-2)(l  - ir)  = 0. 


98 


OF  INTERPRETATION. 


[CHAP.  VI. 


The  independent  relations  here  given  are  the  same  as  we 
before  arrived  at,  as  they  evidently  ought  to  be,  since  whatever 
relations  prevail  independently  of  the  existence  of  a given  class 
of  objects  y,  prevail  independently  also  of  the  existence  of  the  con- 
trary class  1 - y. 

The  direct  solution  afforded  by  the  first  equation  is : — Irra- 
tional persons  consist  of  all  irresponsible  beings  who  are  either  free  to 
act,  or  have  voluntarily  sacrificed  their  liberty,  and  are  not  free  to 
act  ; together  with  an  indefinite  remainder  of  irresponsible  beings 
ivho  have  not  sacrificed  their  liberty,  and  are  not  free  to  act. 

18.  The  propositions  analyzed  in  this  chapter  have  been  of 
that  species  called  definitions.  I have  discussed  none  of  which 
the  second  or  predicate  term  is  particular,  and  of  which  the  ge- 
neral type  is  Y = vX,  Y and  X being  functions  of  the  logical 
symbols  x,  y,  z,  &c.,  and  v an  indefinite  class  symbol.  The  ana- 
lysis of  such  propositions  is  greatly  facilitated  (though  the  step 
is  not  an  essential  one)  by  the  elimination  of  the  symbol  v,  and 
this  process  depends  upon  the  method  of  the  next  chapter.  I 
postpone  also  the  consideration  of  another  important  problem 
necessary  to  complete  the  theory  of  single  propositions,  but  of 
which  the  analysis  really  depends  upon  the  method  of  the  reduc- 
tion of  systems  of  propositions  to  be  developed  in  a future  page 
of  this  work. 


CHAP.  VII.] 


OF  ELIMINATION. 


99 


CHAPTER  VII. 

ON  ELIMINATION. 

1.  TN  the  examples  discussed  in  the  last  chapter,  all  the  ele- 
ments  of  the  original  premiss  re-appeared  in  the  conclusion, 

. only  in  a different  order,  and  with  a different  connexion.  But  it 
more  usually  happens  in  common  reasoning,  and  especially  when 
we  have  more  than  one  premiss,  that  some  of  the  elements  are 
required  not  to  appear  in  the  conclusion.  Such  elements,  or,  as 
they  are  commonly  called,  “ middle  terms,”  may  be  considered 
as  introduced  into  the  original  propositions  only  for  the  sake  of 
that  connexion  which  they  assist  to  establish  among  the  other 
elements,  which  are  alone  designed  to  enter  into  the  expression  of 
the  conclusion. 

2.  Respecting  such  intermediate  elements,  or  middle  terms, 
some  erroneous  notions  prevail.  It  is  a general  opinion,  to  which, 
however,  the  examples  contained  in  the  last  chapter  furnish  a con- 
tradiction, that  inference  consists  peculiarly  in  the  elimination  of 
such  terms,  and  that  the  elementary  type  of  this  process  is  exhi- 
bited in  the  elimination  of  one  middle  term  from  two  premises,  so  as 
to  produce  a single  resulting  conclusion  into  which  that  term  does 
not  enter.  Hence  it  is  commonly  held,  that  syllogism  is  the  basis, 
or  else  the  common  type,  of  all  inference,  which  may  thus,  how- 
ever complex  its  form  and  structure,  be  resolved  into  a series  of 
syllogisms.  The  propriety  of  this  view  will  be  considered  in  a 
subsequent  chapter.  At  present  I wish  to  direct  attention  to  an 
important,  but  hitherto  unnoticed,  point  of  difference  between 
the  system  of  Logic,  as  expressed  by  symbols,  and  that  of  com- 
mon algebra,  with  reference  to  the  subject  of  elimination.  In 
the  algebraic  system  we  are  able  to  eliminate  one  symbol  from 
two  equations,  two  symbols  from  three  equations,  and  generally 
n - 1 symbols  from  n equations.  There  thus  exists  a definite 
connexion  between  the  number  of  independent  equations  given, 


100 


OF  ELIMINATION. 


[CHAP.  VII. 

and  the  number  of  symbols  of  quantity  which  it  is  possible  to 
eliminate  from  them.  But  it  is  otherwise  with  the  system  of 
Logic.  No  fixed  connexion  there  prevails  between  the  num- 
ber of  equations  given  representing  propositions  or  premises, 
and  the  number  of  typical  symbols  of  which  the  elimination 
can  be  effected.  From  a single  equation  an  indefinite  num- 
ber of  such  symbols  may  be  eliminated.  On  the  other  hand, 
from  an  indefinite  number  of  equations,  a single  class  symbol 
only  may  be  eliminated.  We  may  affirm,  that  in  this  peculiar 
system,  the  problem  of  elimination  is  resolvable  under  all  circum- 
stances alike.  This  is  a consequence  of  that  remarkable  law  of 
duality  to  which  the  symbols  of  Logic  are  subject.  To  the  equa- 
tions furnished  by  the  premises  given,  there  is  added  another 
equation  or  system  of  equations  drawn  from  the  fundamental 
laws  of  thought  itself,  and  supplying  the  necessary  means  for  the 
solution  of  the  problem  in  question.  Of  the  many  consequences 
which  flow  from  the  law  of  duality,  this  is  perhaps  the  most 
deserving  of  attention. 

3.  As  in  Algebra  it  often  happens,  that  the  elimination  of 
symbols  from  a given  system  of  equations  conducts  to  a mere 
identity  in  the  form  0 = 0,  no  independent  relations  connecting 
the  symbols  which  remain ; so  in  the  system  of  Logic,  a like  re- 
sult, admitting  of  a similar  interpretation,  may  present  itself. 
Such  a circumstance  does  not  detract  from  the  generality  of 
the  principle  before  stated.  The  object  of  the  method  upon 
which  we  are  about  to  enter  is  to  eliminate  any  number  of  sym- 
bols from  any  number  of  logical  equations,  and  to  exhibit  in  the 
result  the  actual  relations  which  remain.  Now  it  may  be,  that 
no  such  residual  relations  exist.  In  such  a case  the  truth  of  the 
method  is  shown  by  its  leading  us  to  a merely  identical  propo- 
sition. 

4.  The  notation  adopted  in  the  following  Propositions  is 
similar  to  that  of  the  last  chapter.  By  / (x)  is  meant  any  ex- 
pression involving  the  logical  symbol  x,  with  or  without  other 
logical  symbols.  By  /(l)  is  meant  what  f(x)  becomes  when  x 
is  therein  changed  into  1 ; by  /( 0)  what  the  same  function  be- 
comes when  x is  changed  into  0. 


CHAP.  VII.] 


OF  ELIMINATION. 


101 


Proposition  I. 


5.  Iff  {x)  = 0 be  any  logical  equation  involving  the  class  symbol 
x , with  or  without  other  class  symbols,  then  will  the  equation 

fa)  /( o)=o 

be  true,  independently  of  the  interpretation  of  x ; and  it  will  be  the 
complete  result  of  the  elimination  of  x from  the  above  equation. 

In  other  words,  the  elimination  of  x from  any  given  equation, 
f(  x)  = 0,ivill  be  effected  by  successively  changing  in  that  equation  xinto 
1,  and  x into  0,  and  multiplying  the  two  resulting  equations  together. 

Similarly  the  complete  result  of  the  elimination  of  any  class  sym- 
bols, x,  y,  Sfc.,from  any  equation  of  the  form  V=(),  ivill  be  obtained 
by  completely  expanding  the  first  member  of  that  equation  in  con- 
stituents of  the  given  symbols , and  multiplying  together  all  the  coeffi- 
cients of  those  constituents,  and  equating  the  product  to  0. 

Developing  the  first  member  of  the  equation  f(x)  = 0,  we 
have  (V.  10), 

/(l)*+/(0)(l-*)-0; 


or, 


and 


f/O)  -/(°)J  *+/(0)  = 0. 

. . m . 

" m-fw 

i _S____ZQL_ 

m -for 


(i) 


Substitute  these  expressions  for  x and  1 - x in  the  fundamental 
equation 

x (1  - x)  = 0, 

and  there  results 


mm  0. 

~ l/(0)-/(l))2_U’ 

or,  /(l)/(0)  = 0,  (2) 

the  form  required. 

6.  It  is  seen  in  this  process,  that  the  elimination  is  really  effected 
between  the  given  equation  f(x)  = 0 and  the  universally  true 
equation  x (1  - x)  = 0,  expressing  the  fundamental  law  of  logical 
symbols,  qua  logical.  There  exists,  therefore,  no  need  of  more 


102 


OF  ELIMINATION. 


[CHAP.  VII. 


than  one  premiss  or  equation,  in  order  to  render  possible  the  eli- 
mination of  a term,  the  necessary  law  of  thought  virtually  sup- 
plying  the  other  premiss  or  equation.  And  though  the  demon- 
stration of  this  conclusion  may  be  exhibited  in  other  forms,  yet 
the  same  element  furnished  by  the  mind  itself  will  still  be  vir- 
tually present.  Thus  we  might  proceed  as  follows  : 

Multiply  (1)  by  a;,  and  we  have 

/ (1)  * = 0,  (3) 

and  let  us  seek  by  the  forms  of  ordinary  algebra  to  eliminate  x 
from  this  equation  and  (1). 

Now  if  we  have  two  algebraic  equations  of  the  form 

ax  + b = 0, 
a'x  + b'  = 0 ; 

it  is  well  known  that  the  result  of  the  elimination  of  x is 

ab’  - ab  - 0.  (4) 

But  comparing  the  above  pair  of  equations  with  (1)  and  (3) 
respectively,  we  find 

WO)  -/(«)’  W(»): 

WO) 

which,  substituted  in  (4),  give 

/(l)/(0)-0, 

as  before.  In  this  form  of  the  demonstration,  the  fundamental 
equation  *(1  - x)  = 0,  makes  its  appearance  in  the  derivation  of 
(3)  from  (1). 

7.  I shall  add  yet  another  form  of  the  demonstration,  par- 
taking of  a half  logical  character,  and  which  may  set  the  demon- 
stration of  this  important  theorem  in  a clearer  light. 

We  have  as  before 

/(l)  x +/(°)(1  - #)  = 0. 

Multiply  this  equation  first  by  x , and  secondly  by  1 - x,  we  get 
/(1)*  = 0,  /(0) (1  - x)  = 0. 

From  these  we  have  by  solution  and  development, 


CHAP.  VII.] 


OF  ELIMINATION. 


103 


/(1)  = - = ^(1-#),  on  development, 

CC  l> 


0 0 


The  direct  interpretation  of  these  equations  is — 

1st.  Whatever  individuals  are  included  in  the  class  repre- 
sented by  /(l),  are  not  x’s. 

2nd.  Whatever  individuals  are  included  in  the  class  repre- 
sented by/(0),  are  a?’s. 

Whence  by  common  logic,  there  are  no  individuals  at  once 
in  the  class  /(l)  and  in  the  class  / (0),  i.e.  there  are  no  indivi- 
duals in  the  class/ (1)  /( 0).  Hence, 

/(l)/(0)-0.  (5) 

Or  it  would  suffice  to  multiply  together  the  developed  equa- 
tions, whence  the  result  would  immediately  follow. 

8.  The  theorem  (5)  furnishes  us  with  the  following  Rule : 

TO  ELIMINATE  ANY  SYMBOL  FROM  A PROPOSED  EQUATION. 

Rule. — The  terms  of  the  equation  having  been  brought , by  trans- 
position if  necessary , to  the  first  side,  give  to  the  symbol  successively 
the  values  1 and  0,  and  multiply  the  resulting  equations  together. 

The  first  part  of  the  Proposition  is  now  proved. 

9.  Consider  in  the  next  place  the  general  equation 

/O,  y)  = 0; 

the  first  member  of  which  represents  any  function  of  x,  y,  and 
other  symbols. 

By  what  has  been  shown,  the  result  of  the  elimination  of  y 
from  this  equation  will  be 

f(x,  1 )/(>,  0)  = 0 ; 

for  such  is  the  form  to  which  we  are  conducted  by  successively 
changing  in  the  given  equation  y into  1,  and  y into  0,  and  multi- 
plying the  results  together. 

Again,  if  in  the  result  obtained  we  change  successively  a:  into 
1,  and  x into  0,  and  multiply  the  results  together,  we  have 
/(1,1)/(1,0)/(G,  1)  / (0,  0)  --=  0 ; 
as  the  final  result  of  elimination. 


(6) 


104 


OF  ELIMINATION. 


[CHAP.  VII. 

But  the  four  factors  of  the  first  member  of  this  equation  are 
the  four  coefficients  of  the  complete  expansion  of  / ( x , y),  the 
first  member  of  the  original  equation ; whence  the  second  part  of 
the  Proposition  is  manifest. 

EXAMPLES. 

10.  Ex.  1. — Having  given  the  Proposition,  “All  men  are 
mortal,”  and  its  symbolical  expression,  in  the  equation, 

y = vx, 

in  which  y represents  “ men,”  and  x “ mortals,”  it  is  required  to 
eliminate  the  indefinite  class  symbol  v , and  to  interpret  the 
result. 

Here  bringing  the  terms  to  the  first  side,  we  have 
y - vx  = 0. 

When  v = 1 this  becomes 

y-x  = 0; 

and  when  v = 0 it  becomes 

y = 0; 

and  these  two  equations  multiplied  together,  give 

y-yx  = 0, 

or  y (1  - x)  = 0, 

it  being  observed  that  y 2 = y. 

The  above  equation  is  the  required  result  of  elimination,  and 
its  interpretation  is,  Men  who  are  not  mortal  do  not  exist , — an 
obvious  conclusion. 

If  from  the  equation  last  obtained  we  seek  a description  of 
beings  who  are  not  mortal,  we  have 


• • 1 ” X — • 

y 

Whence,  by  expansion,  1 - x = ^ ( 1 - y),  which  interpreted  gives, 
They  who  are  not  mortal  are  not  men.  This  is  an  example  of 


OF  ELIMINATION. 


105 


CHAP.  VII.] 


what  in  the  common  logic  is  called  conversion  by  contraposition, 
or  negative  conversion.* 

Ex.  2. — Taking  the  Proposition,  “ No  men  are  perfect,”  as 
represented  by  the  equation 

y = v{\  - x), 

wherein  y represents  “ men,”  and  x “ perfect  beings,”  it  is  re- 
quired to  eliminate  v,  and  find  from  the  result  a description  both 
of  perfect  beings  and  of  imperfect  beings.  W e have 
y - v { 1 - x)  = 0. 

Whence,  by  the  rule  of  elimination, 

{y  ~ (i  -*)}  x y = o, 

or  y - y (1  - x)  = 0, 

or  yx  = 0 ; 


which  is  interpreted  by  the  Proposition,  Perfect  men  do  not  exist. 
From  the  above  equation  we  have 


0 

x - - 

y 


= - (1  - y)  by  development; 


whence,  by  interpretation,  No  perfect  beings  are  men. 
larly, 


, . 0 w 0 N 

"y"i'!'+oC  y)' 


Simi- 


which,  on  interpretation,  gives,  Imperfect  beings  are  all  men  with 
an  indefinite  remainder  of  beings,  which  are  not  men. 

11.  It  will  generally  be  the  most  convenient  course,  in  the 
treatment  of  propositions,  to  eliminate  first  the  indefinite  class 
symbol  v,  wherever  it  occurs  in  the  corresponding  equations. 
This  will  only  modify  their  form,  without  impairing  their  signifi- 
cance. Let  us  apply  this  process  to  one  of  the  examples  of 
Chap.  iv.  For  the  Proposition,  “ No  men  are  placed  in  exalted 
stations  and  free  from  envious  regards,”  we  found  the  expression 


y = v (1  - xz ), 

and  for  the  equivalent  Proposition,  “ Men  in  exalted  stations  are 
not  free  from  envious  regards,”  the  expression 
yx  = w(l  - z); 


Whately’s  Logic,  Book  II.  chap.  II.  sec.  4. 


106 


OF  ELIMINATION. 


[CHAP.  VII. 

and  it  was  observed  that  these  equations,  v being  an  indefinite 
class  symbol,  were  themselves  equivalent.  To  prove  this,  it  is 
only  necessary  to  eliminate  from  each  the  symbol  v.  The  first 
equation  is 

y - v (1  - xz ) = 0, 

whence,  first  making  v = 1 , and  then  v = 0,  and  multiplying  the 
results,  we  have 

(y  - 1 + xz)  y = 0, 

Or  yxz  = 0. 

Now  the  second  of  the  given  equations  becomes  on  transposition 
yx  - v (1  - z)  = 0 ; 
whence  (yx  - 1 + z)  yx  = 0, 

or  yxz  = 0, 

as  before.  The  reader  will  easily  interpret  the  result. 

12.  Ex.  3. — As  a subject  for  the  general  method  of  this 
chapter,  we  will  resume  Mr.  Senior’s  definition  of  wealth,  viz. : 
“ Wealth  consists  of  things  transferable,  limited  in  supply,  and 
either  productive  of  pleasure  or  preventive  of  pain.”  We  shall 
consider  this  definition,  agreeably  to  a former  remark,  as  including 
all  things  which  possess  at  once  both  the  qualities  expressed  in 
the  last  part  of  the  definition,  upon  which  assumption  we  have, 
as  our  representative  equation, 


tv  = st  [pr  + p ( 1 - r)  + r (1  - p) } , 

or 

tv  = 

st  {p  + r(  1 -jo)), 

wherein 

w stands  for  wealth. 

things  limited  in  supply. 

things  transferable. 

P 

things  productive  of  pleasure. 

r „ 

things  preventive  of  pain. 

From  the  above  equation  we  can  eliminate  any  symbols  that 
we  do  not  desire  to  take  into  account,  and  express  the  result  by 
solution  and  development,  according  to  any  proposed  arrange- 
ment of  subject  and  predicate. 

Let  us  first  consider  what  the  expression  lor  tv,  wealth,  would 


OF  ELIMINATION. 


107 


CHAP.  VII.] 

be  If  the  element  r,  referring  to  prevention  of  pain,  were  elimi- 
nated. Now  bringing  the  terms  of  the  equation  to  the  first  side, 
we  get 

w - st  (p  + r - rp)  = 0. 

Making  r = 1,  the  first  member  becomes  w - st,  and  making 
r = 0 it  becomes  w - stp ; whence  we  have  by  the  Ride, 

(w  - st)  (w  - stp)  = 0,  (7) 

or  to  - wstp  - wst  + stp  = 0 ; (8) 

whence  stp 

w = - ; 

st  + stp  - 1 

the  development  of  the  second  member  of  which  equation  gives 


w 


= stp  + - st  (1  - p).] 


(9) 


Whence  we  have  the  conclusion, — Wealth  consists  of  all  things 
limited  in  supply , transferable,  and  productive  of  pleasure,  and  an 
indefinite  remainder  of  things  limited  in  supply,  transferable,  and 
not  productive  of  pleasure.  This  is  sufficiently  obvious. 

Let  it  be  remarked  that  it  is  not  necessary  to  perform  the 
multiplication  indicated  in  (7 ),  and  reduce  that  equation  to  the 
form  (8),  in  order  to  determine  the  expression  of  ic  in  terms  of 
the  other  symbols.  The  process  of  development  may  in  all  cases 
be  made  to  supersede  that  of  multiplication.  Thus  if  we  de- 
velop (7)  in  terms  of  w,  we  find 


whence 


(1  - st)  (1  - stp)  iv  + stp  (1  - w)  = 0, 
stp 


to  = 


stp  - (1  - st)  (1  - stp)  s 
and  this  equation  developed  will  give,  as  before, 

w = stp  + - st  (1  - p). 

13.  Suppose  next  that  we  seek  a description  of  things  limited 
in  supply,  as  dependent  upon  their  relation  to  wealth,  transferable- 
ness, and  tendency  to  produce  pleasure,  omitting  all  reference  to 
the  prevention  of  pain. 


108  OF  ELIMINATION.  [CHAP.  VII. 

From  equation  (8),  which  is  the  result  of  the  elimination  of 
t from  the  original  equation,  we  have 

w - s (wt  + wtp  - tp)  - 0 ; 

whence  w 

s = 

wt  + wtp  - tp 

= ivlp  + wt  (1  -p)  + i w (1  - t)p  + (1  - t)  (1  - p) 

+ 0 (1  -w)tp  + jj(l  - w)  t (1  -p)  + ^(l  - w)  (1  - t)p 

+ jjo  ~w)  C1  -0  0 “ P )• 

We  will  first  give  the  direct  interpretation  of  the  above  solution, 
term  by  term ; afterwards  we  shall  offer  some  general  remarks 
which  it  suggests ; and,  finally,  show  how  the  expression  of  the 
conclusion  may  be  somewhat  abbreviated. 

First,  then,  the  direct  interpretation  is,  Things  limited  in 
supply  consist  of  All  wealth  transferable  and  productive  of  pleasure 
— all  wealth  transferable,  and  not  productive  of  pleasure, — an  indefi- 
nite amount  of  what  is  not  wealth,  but  is  either  transferable,  and  not 
productive  of  pleasure,  or  intransfer  able  and  productive  of  pleasure, 
or  neither  transferable  nor  productive  of  pleasure. 

To  which  the  terms  whose  coefficients  are  ^ permit  us  to  add 
the  following  independent  relations,  viz. : 

1st.  Wealth  that  is  intransfer  able,  and  productive  of  pleasure, 
does  not  exist. 

2ndly.  Wealth  that  is  intransf enable , and  not  productive  of  plea- 
sure, does  not  exist. 

14.  Respecting  this  solution  I suppose  the  following  remarks 
are  likely  to  be  made. 

First,  it  may  be  said,  that  in  the  expression  above  obtained 
for  “ things  limited  in  supply,”  the  term  “ All  wealth  transfer- 
able,” &c.,  is  in  part  redundant ; since  all  wealth  is  (as  implied 
in  the  original  proposition,  and  directly  asserted  in  the  indepen- 
dent relations ) necessarily  transferable. 

I answer,  that  although  in  ordinary  speech  we  should  not 


CHAP.  VII.] 


OF  ELIMINATION. 


109 


deem  it  necessary  to  add  to  “wealth”  the  epithet  “ transferable,” 
if  another  part  of  our  reasoning  had  led  us  to  express  the  con- 
clusion, that  there  is  no  wealth  which  is  not  transferable,  yet  it 
pertains  to  the  perfection  of  this  method  that  it  in  ail  cases  fully 
defines  the  objects  represented  by  each  term  of  the  conclusion, 
by  stating  the  relation  they  bear  to  each  quality  or  element  of  dis- 
tinction that  we  have  chosen  to  employ.  This  is  necessary  in  order 
to  keep  the  different  parts  of  the  solution  really  distinct  and  in- 
dependent, and  actually  prevents  redundancy.  Suppose  that  the 
pair  of  terms  we  have  been  considering  had  not  contained  the 
word  “ transferable,”  and  had  unitedly  been  “All  wealth,”  we 
could  then  logically  resolve  the  single  term  “ All  wealth”  into 
the  two  terms  “ All  wealth  transferable,”  and  “ All  wealth 
intransferable.”  But  the  latter  term  is  shown  to  disappear  by 
the  “independent  relations.”  Hence  it  forms  no  part  of  the  de- 
scription required,  and  is  therefore  redundant.  The  remaining 
term  agrees  with  the  conclusion  actually  obtained. 

Solutions  in  which  there  cannot,  by  logical  divisions,  be  pro- 
duced any  superfluous  or  redundant  terms,  may  be  termed  pure 
solutions.  Such  are  all  the  solutions  obtained  by  the  method  of 
development  and  elimination  above  explained.  It  is  proper  to 
notice,  that  if  the  common  algebraic  method  of  elimination  were 
adopted  in  the  cases  in  which  that  method  is  possible  in  the  pre- 
sent system,  we  should  not  be  able  to  depend  upon  the  purity  of 
the  solutions  obtained.  Its  want  of  generality  would  not  be  its 
only  defect. 

15.  In  the  second  place,  it  will  be  remarked,  that  the  con- 
clusion contains  two  terms,  the  aggregate  significance  of  which 
would  be  more  conveniently  expressed  by  a single  term.  Instead 
of  “ All  wealth  productive  of  pleasure,  and  transferable,”  and 
“All  wealth  not  productive  of  pleasure,  and  transferable,”  we 
might  simply  say,  “ All  wealth  transferable.”  This  remark  is 
quite  just.  But  it  must  be  noticed  that  whenever  any  such  sim- 
plifications are  possible,  they  are  immediately  suggested  by  the 
form  of  the  equation  we  have  to  interpret ; and  if  that  equation 
be  reduced  to  its  simplest  form,  then  the  interpretation  to  which 
it  conducts  will  be  in  its  simplest  form  also.  Thus  in  the  original 
solution  the  terms  wtp  and  wt  ( 1 - p),  which  have  unity  for  their 


OF  ELIMINATION. 


110 


[chap.  VII. 


coefficient,  give,  on  addition,  wt ; the  terms  w (1  - t)  p and 
( 1 - t)  ( 1 - p),  which  have  - for  their  coefficient  give  to  ( 1 - t) ; 
and  the  terms  (1  - w)  (1  - t)p  and  (1  - w ) (1  - 1 ) (1  -p),  which 
have  - for  their  coefficient,  give  (1  - w)  (1  - t).  Whence  the 
complete  solution  is 

« = wt  + C1  _ w)  (1 0 + ^ (1  “ w)  t C1  “ P)> 

with  the  independent  relation, 

0 

w ( 1 - t)  = 0,  or  w = -f. 

The  interpretation  would  now  stand  thu3  : — 

1st.  Things  limited  in  supply  consist  of  all  wealth  transferable , 
with  an  indefinite  remainder  of  ichat  is  not  wealth  and  not  transfer- 
able, and  of  transferable  articles  which  are  not  wealth , and  are  not 
productive  of  pleasure. 

2nd.  All  wealth  is  transferable. 

This  is  the  simplest  form  under  which  the  general  conclusion, 
with  its  attendant  condition,  can  be  put. 

16.  When  it  is  required  to  eliminate  two  or  more  symbols 
from  a proposed  equation  we  can  either  employ  (6)  Prop.  I.,  or 
eliminate  them  in  succession,  the  order  of  the  process  being  in- 
different. From  the  equation 

w = st(p  + r - pr ), 
we  have  eliminated  r,  and  found  the  result, 

w - wst  - wstp  + stp  = 0. 

Suppose  that  it  had  been  required  to  eliminate  both  r and  t,  then 
taking  the  above  as  the  first  step  of  the  process,  it  remains  to 
eliminate  from  the  last  equation  t.  Now  when  t=  1 the  first 
member  of  that  equation  becomes 

w - tos  - ivsp  + sp, 

and  when  t = 0 the  same  member  becomes  iv.  Whence  we  have 
w ( w - ivs  - wsp  + sp)  = 0, 
or  w - ws  = 0, 

for  the  required  result  of  elimination. 


CHAP.  VII.] 


OF  ELIMINATION. 


Ill 


If  from  the  last  result  we  determine  w,  we  have 

0 0 

w = - = - 5, 

1 — s 0 

whence  “ All  wealth  is  limited  in  supply.”  As  p does  not  enter 
into  the  equation,  it  is  evident  that  the  above  is  true,  irrespec- 
tively of  any  relation  which  the  elements  of  the  conclusion  bear 
to  the  quality  “ productive  of  pleasure.” 

Resuming  the  original  equation,  let  it  be  required  to  elimi- 
nate s and  t.  We  have 

w = st  (p  + r - pr ). 

Instead,  however,  of  separately  eliminating  s and  t according  to 
the  Rule,  it  will  suffice  to  treat  st  as  a single  symbol,  seeing  that 
it  satisfies  the  fundamental  law  of  the  symbols  by  the  equation 

st  (1  - st)  = 0. 

Placing,  therefore,  the  given  equation  under  the  form 
w - st  (p  + r - pr)  = 0 ; 

and  making  st  successively  equal  to  1 and  to  0,  and  taking  the 
product  of  the  results,  we  have 

(w  - p - r + pr)  w - 0, 

or  w - wp  - wr  + wpr  - 0, 

for  the  result  sought. 

As  a particular  illustration,  let  it  be  required  to  deduce  an 
expression  for  “ things  productive  of  pleasure”  (p),  in  terms  of 
“ wealth”  (ic),  and  “ things  preventive  of  pain”  (r)\ 

We  have,  on  solving  the  equation, 

w ( 1 - r) 

^ in  (l  - r) 

= ^w  + ?/r(l  _ r)  + (l  “ w)  r + ^ (1  ~ w)  (1  ~ r) 

= w (1  - r)  + ^ wr  + (1  - to). 

Whence  the  following  conclusion: — Things  productive  of  plea- 


112 


OF  ELIMINATION. 


[C 


HAP.  VII. 


sure  are,  all  wealth  not  preventive  of  pain,  an  indefinite  amount 
of  wealth  that  is  preventive  of  pain,  and  an  indefinite  amount  of 
what  is  not  wealth. 

From  the  same  equation  we  get 


i | »0-0_  ° 

w (1  - r)  w (1  - rf 

which  developed,  gives 

w (1  ~P)  = ^ wr  + ^ 0 “ w)  • T + ^ (1  ~ w)  • (1  - r) 

0 0 

'ow  + o 

Whence,  Things  not  productive  of  pleasure  are  either  wealth , pre- 
ventive of  pain,  or  what  is  not  wealth. 

Equally  easy  would  be  the  discussion  of  any  similar  case. 

17.  In  the  last  example  of  elimination,  we  have  eliminated 
the  compound  symbol  st  from  the  given  equation,  by  treating  it 
as  a single  symbol.  The  same  method  is  applicable  to  any  com- 
bination of  symbols  which  satisfies  the  fundamental  law  of  indi- 
vidual symbols.  Thus  the  expression  p + r - pr  will,  on  being 
multiplied  by  itself,  reproduce  itself,  so  that  if  we  represent 
p + r - pr  by  a single  symbol  as  y,  we  shall  have  the  fundamen- 
tal law  obeyed,  the  equation 

V = y\  or  y (l  -y)  = 0, 


O - w )• 


being  satisfied.  F or  the  rule  of  elimination  for  symbols  is  founded 
upon  the  supposition  that  each  individual  symbol  is  subject  to 
that  law  ; and  hence  the  elimination  of  any  function  or  combina- 
tion of  such  symbols  from  an  equation,  may  be  effected  by  a sin- 
gle operation,  whenever  that  law  is  satisfied  by  the  function. 

Though  the  forms  of  interpretation  adopted  in  this  and  the 
previous  chapter  show,  perhaps  better  than  any  others,  the  di- 


rect significance  of  the  symbols  1 and 


0 

0* 


modes  of  expression 


more  agreeable  to  those  of  common  discourse  may,  with  equal 
truth  and  propriety,  be  employed.  Thus  the  equation  (9)  may 
be  interpreted  in  the  following  manner : Wealth  is  either  limited 
in  supply,  transferable,  and  productive  of  pleasure,  or  limited  in  sup- 


CHAP.  VII.] 


OF  ELIMINATION. 


113 


ply,  transferable , and  not  productive  of  pleasure.  And  reversely, 
Whatever  is  limited  in  supply,  transferable , and  productive  of  plea- 
sure, is  wealth.  Reverse  interpretations,  similar  to  the  above,  are 
always  furnished  when  the  final  development  introduces  terms 
having  unity  as  a coefficient. 

18.  Note. — The  fundamental  equation  /(l)/(0)  = 0,  ex- 
pressing the  result  of  the  elimination  of  the  symbol  x from  any 
equation  f (.r)  = 0,  admits  of  a remarkable  interpretation. 

It  is  to  be  remembered,  that  by  the  equation /(x)  = 0 is  im- 
plied some  proposition  in  which  the  individuals  represented  by 
the  class  x,  suppose  “ men,”  are  referred  to,  together,  it  may  be, 
with  other  individuals ; and  it  is  our  object  to  ascertain  whether 
there  is  implied  in  the  proposition  any  relation  among  the  other 
individuals,  independently  of  those  found  in  the  class  men.  Now 
the  equation  /(l)  = 0 expresses  what  the  original  proposition 
would  become  if  men  made  up  the  universe,  and  the  equation 
/(0)  = 0 expresses  what  that  original  proposition  would  become 
if  men  ceased  to  exist,  wherefore  the  equation  /( 1)  /( 0)  = 0 ex- 
presses what  in  virtue  of  the  original  proposition  would  be 
equally  true  on  either  assumption,  i.  e.  equally  true  whether 
“men”  were  “all  things”  or  “nothing.”  Wherefore  the  theo- 
rem expresses  that  what  is  equally  true,  whether  a given  class  of 
objects  embraces  the  whole  universe  or  disappears  from  existence, 
is  independent  of  that  class  altogether,  and  vice  versa.  Herein 
we  see  another  example  of  the  interpretation  of  formal  results, 
immediately  deduced  from  the  mathematical  laws  of  thought,  into 
general  axioms  of  philosophy. 


114 


OF  REDUCTION. 


[CHAP.  VIII. 


CHAPTER  VIII. 

ON  THE  REDUCTION  OF  SYSTEMS  OF  PROPOSITIONS. 

E TN  the  preceding  chapters  we  have  determined  sufficiently 
for  the  most  essential  purposes  the  theory  of  single  pri- 
mary propositions,  or,  to  speak  more  accurately,  of  primary  pro- 
positions expressed  by  a single  equation.  And  we  have  estab- 
lished upon  that  theory  an  adequate  method.  We  have  shown 
how  any  element  involved  in  the  given  system  of  equations  may 
be  eliminated,  and  the  relation  which  connects  the  remaining 
elements  deduced  in  any  proposed  form,  whether  of  denial,  of  af- 
firmation, or  of  the  more  usual  relation  of  subject  and  predicate. 
It  remains  that  we  proceed  to  the  consideration  of  systems  of 
propositions,  and  institute  with  respect  to  them  a similar  series 
of  investigations.  We  are  to  inquire  whether  it  is  possible  from 
the  equations  by  which  a system  of  propositions  is  expressed  to 
eliminate,  ad  libitum,  any  number  of  the  symbols  involved ; to 
deduce  by  interpretation  of  the  result  the  whole  of  the  relations 
implied  among  the  remaining  symbols  ; and  to  determine  in  par- 
ticular the  expression  of  any  single  element,  or  of  any  inter- 
pretable combination  of  elements,  in  terms  of  the  other  elements, 
so  as  to  present  the  conclusion  in  any  admissible  form  that  may 
be  required.  These  questions  will  be  answered  by  showing  that  it 
is  possible  to  reduce  any  system  of  equations,  or  any  of  the  equa- 
tions involved  in  a system,  to  an  equivalent  single  equation,  to 
which  the  methods  of  the  previous  chapters  may  be  immediately 
applied.  It  will  be  seen  also,  that  in  this  reduction  is  involved 
an  important  extension  of  the  theory  of  single  propositions,  which 
in  the  previous  discussion  of  the  subject  we  were  compelled  to 
forego.  This  circumstance  is  not  peculiar  in  its  nature.  There 
are  many  special  departments  of  science  which  cannot  be  com- 
pletely surveyed  from  within,  but  require  to  be  studied  also  from 
an  external  point  of  view,  and  to  be  regarded  in  connexion  with 


OF  REDUCTION. 


115 


CHAP.  VIII.] 

other  and  kindred  subjects,  in  order  that  then  full  proportions 
may  be  understood. 

This  chapter  will  exhibit  two  distinct  modes  of  reducing 
systems  of  equations  to  equivalent  single  equations.  The  first 
of  these  rests  upon  the  employment  of  arbitrary  constant  multi- 
pliers. It  is  a method  sufficiently  simple  in  theory,  but  it  has  the 
inconvenience  of  rendering  the  subsequent  processes  of  elimina- 
tion and  development,  when  they  occur,  somewhat  tedious.  It  was, 
however,  the  method  of  reduction  first  discovered,  and  partly  on 
this  account,  and  partly  on  account  of  its  simplicity,  it  has  been 
thought  proper  to  retain  it.  The  second  method  does  not  re- 
quire the  introduction  of  arbitrary  constants,  and  is  in  nearly 
all  respects  preferable  to  the  preceding  one.  It  will,  therefore, 
generally  be  adopted  in  the  subsequent  investigations  of  this 
work. 

2.  We  proceed  to  the  consideration  of  the  first  method. 

Proposition  I. 

Any  system  of  logical  equations  may  be  reduced  to  a single  equiva- 
lent equation , by  multiplying  each  equation  after  the  first  by  a dis- 
tinct arbitrary  constant  quantity , and  adding  all  the  results,  including 
the  first  equation,  together . 

By  Prop.  2,  Chap,  vi.,  the  interpretation  of  any  single 
equation,  f(x,  y . .)  = 0 is  obtained  by  equating  to  0 those  con- 
stituents of  the  development  of  the  first  member,  whose  co- 
efficients do  not  vanish.  And  hence,  if  there  be  given  two  equa- 
tions, f(x,y..)  = 0,  and  F(x,  y . .)  = 0,  their  united  import  will  be 
contained  in  the  system  of  results  formed  by  equating  to  0 all 
those  constituents  which  thus  present  themselves  in  both,  or  in 
either,  of  the  given  equations  developed  according  to  the  Pule  of 
Chap.  vi.  Thus  let  it  be  supposed,  that  we  have  the  two  equations 

xy  - 2x  = 0,  ( 1 ) 

x~y  = 0 ; (2) 

The  development  of  the  first  gives 

- xy  - 2x  (1  - y)  = 0 ; 
xy  = 0,  x(l-y)  = 0. 

i 2 


whence, 


(3) 


116 


OF  REDUCTION. 


[CHAP.  VIII. 


The  development  of  the  second  equation  gives 

x 0 - y)  - y 0 - x)  = 0 ; 

whence,  x (1  - y)  = 0,  y (1  - x)  = 0.  (4) 

The  constituents  whose  coefficients  do  not  vanish  in  both  deve- 
lopments are  xy,  x (1  - y),  and  (1  - x)y,  and  these  would  to- 
gether give  the  system 

xy  = 0,  x (1  - y)  = 0,  (1  - x)  y = 0 ; (5) 

which  is  equivalent  to  the  two  systems  given  by  the  developments 
separately,  seeing  that  in  those  systems  the  equation  x (1  - y)  = 0 
is  repeated.  Confining  ourselves  to  the  case  of  binary  systems 
of  equations,  it  remains  then  to  determine  a single  equation, 
which  on  development  shall  yield  the  same  constituents  with 
coefficients  which  do  not  vanish,  as  the  given  equations  produce. 
Now  if  we  represent  by 

V1  = 0,  V2  = 0, 

the  given  equations,  F,  and  V2  being  functions  of  the  logical  sym- 
bols x,  y,  z,  &c. ; then  the  single  equation 

V\  + cV2=  0,  (6) 

c being  an  arbitrary  constant  quantity,  will  accomplish  the  re- 
quired object.  For  let  At  represent  any  term  in  the  full  de- 
velopment Fj  wherein  i is  a constituent  and  A its  numerical 
coefficient,  and  let  Bt  represent  the  corresponding  term  in  the 
full  development  of  V2,  then  will  the  corresponding  term  in  the 
development  of  (6)  be 

(A  + cB)  t. 

The  coefficient  of  t vanishes  if  A and  B both  vanish,  but  not 
otherwise.  For  if  we  assume  that  A and  B do  not  both  vanish, 
and  at  the  same  time  make 

A + cB  = 0,  (7) 

the  following  cases  alone  can  present  themselves. 

1st.  That  A vanishes  and  B does  not  vanish.  In  this  case 
the  above  equation  becomes 

cB  = 0, 


OF  REDUCTION. 


117 


CHAF.  VIII.] 

and  requires  that  c = 0.  But  this  contradicts  the  hypothesis  that 
c is  an  arbitrary  constant. 

2nd.  That  B vanishes  and  A does  not  vanish.  This  assump- 
tion reduces  (7)  to 

A = 0, 

by  which  the  assumption  is  itself  violated. 

3rd.  That  neither  A nor  B vanishes.  The  equation  (7)  then 
gives 

- A 


which  is  a definite  value,  and,  therefore,  conflicts  with  the  hy- 
pothesis that  c is  arbitrary. 

Hence  the  coefficient  A + cB  vanishes  when  A and  B both 
vanish,  but  not  otherwise.  Therefore,  the  same  constituents 
will  appear  in  the  development  of  (6),  with  coefficients  which  do 
not  vanish,  as  in  the  equations  Vl  =0,  V2  = 0,  singly  or  together. 
And  the  equation  Vx  + cV2=  0,  will  be  equivalent  to  the  sys- 
tem Fi  = 0,  F2  = 0. 

By  similar  reasoning  it  appears,  that  the  general  system  of 
equations 

Fj  = 0,  F2  = 0,  V3  = 0,  &c. ; 
may  be  replaced  by  the  single  equation 

F,  + cF2  + c'F3  + &c.  = 0, 

c,  c,  &c.,  being  arbitrary  constants.  The  equation  thus  formed 
may  be  treated  in  all  respects  as  the  ordinary  logical  equations 
of  the  previous  chapters.  The  arbitrary  constants  cl5  c2,  &c.,  are 
not  logical  symbols.  They  do  not  satisfy  the  law, 

Ci  (1  - Cj)  = 0,  c2  (1  - c2)  = 0. 

But  their  introduction  is  justified  by  that  general  principle  which 
has  been  stated  in  (II.  15)  and  (V.  6),  and  exemplified  in  nearly 
all  our  subsequent  investigations,  viz.,  that  equations  involving 
the  symbols  of  Logic  may  be  treated  in  all  respects  as  if  those 
symbols  were  symbols  of  quantity,  subject  to  the  special  law 
x (1  - x)  = 0,  until  in  the  final  stage  of  solution  they  assume  a 
form  interpretable  in  that  system  of  thought  with  which  Logic 
is  conversant. 


118  OF  REDUCTION,  [cHAF.VIII. 

3.  The  following  example  will  serve  to  illustrate  the  above 
method. 

Ex.  1. — Suppose  that  an  analysis  of  the  properties  of  a parti- 
cular class  of  substances  has  led  to  the  following  general  conclu- 
sions, viz.  : 

1st.  That  wherever  the  properties  A and  B are  combined, 
either  the  property  C.  or  the  property  1),  is  present  also ; but 
they  are  not  jointly  present. 

2nd,  That  wherever  the  properties  B and  C are  combined, 
the  properties  A and  D are  either  both  present  with  them,  or 
both  absent. 

3rd.  That  wherever  the  properties  A and  B are  both  absent, 
the  properties  C and  D are  both  absent  also;  and  vice  versa , where 
the  properties  C and  I)  are  both  absent,  A and  B are  both  absent 
also. 

Let  it  then  be  required  from  the  above  to  determine  what 
may  be  concluded  in  any  particular  instance  from  the  presence  of 
the  property  A with  respect  to  the  presence  or  absence  of  the 
properties  B and  (7,  paying  no  regard  to  the  property  D. 

Represent  the  property  A by  x ; 

„ the  property  B by  y ; 

„ the  property  C by  z : 

,,  the  property  B by  w. 

Then  the  symbolical  expression  of  the  premises  will  be 

xy  = v [w  (1  - z)  + z (1  - w) j ; 
yz  = v [xw  + (1  - x ) (1  - w) } ; 

(1  - x)  (1  - y)  = (1  - z)  (1  - w). 

From  the  first  two  of  these  equations,  separately  eliminating  the 
indefinite  class  symbol  v,  we  have 

xy  {1  - w (1  - z)  - z (1  - w)j  = 0 ; 
yz  {1  - xw  - (1  - x)(l  - w )}  = 0. 

Now  if  we  observe  that  by  development 

1 — w (1  — z)  - z (1  - w)  = wz  + (1  - w)  (1  - z), 

1 - xw  - (1  - x)  ( 1 - tv)  = x (1  - u>)  + iv  (1  - r). 


and 


CHAP.  VIII.]  OF  REDUCTION.  119 

and  in  these  expressions  replace,  for  simplicity, 

1 - x by  x,  l-ybjy,  &c., 
we  shall  have  from  the  three  last  equations, 

xy  (wz  + wz)  - 0 ; (1) 

yz  (xw  + xw)  = 0 ; (2) 

xy  = wz;  (3) 

and  from  this  system  we  must  eliminate  w. 

Multiplying  the  second  of  the  above  equations  by  c,  and  the 
third  by  d,  and  adding  the  results  to  the  first,  we  have 

xy  (wz  + wz)  + cyz  (xw  + xw)  + c' (xy  - wz)  = 0. 

When  w is  made  equal  to  1,  and  therefore  w to  0,  the  first  mem- 
ber of  the  above  equation  becomes 

xyz  + cxyz  + dry. 

And  when  in  the  same  member  w is  made  0 and  w = 1,  it  be- 
comes 

xyz  + cxyz  + dry  - c'z. 

Hence  the  result  of  the  elimination  of  w may  be  expressed  in  the 
form 

(xyz  + cxyz  + cxy)  (xyz  + cxyz  + cxy  - cz)  = 0 ; (4) 

and  from  this  equation  x is  to  be  determined. 

Were  we  now  to  proceed  as  in  former  instances,  we  should 
multiply  together  the  factors  in  the  first  member  of  the  above 
equation  ; but  it  may  be  well  to  show  that  such  a course  is  not 
at  all  necessary.  Let  us  develop  the  first  member  of  (4)  with 
reference  to  x,  the  symbol  whose  expression  is  sought,  we  find 

yz  (yz  + cyz  - cz)  x + (cyz  + c'y)  (c'y  - c'z)  (1  - x)  = 0 ; 

or,  cyzx  + (cyz  + c'y)  (cy  - cz)  (1  - x)  - 0 ; 

whence  we  find, 

(gyg  + <fy)  (c'y  - 

(cyz  + c'y)  (dy  - c'z)  - cyz  ’ 

and  developing  the  second  member  with  respect  to  y and  2, 


120 


OF  REDUCTION. 


[chap.  VIII. 


„ 0 _ c'2  _ 0 . _ 

®=(l-y)*+jj^l-s)>  jj(l-y)(l-z); 

x = {\-y)z  + ^(l-z)-, 

the  interpretation  of  which  is,  Wherever  the  property  A is  present, 
there  either  C is  present  and  B absent,  or  C is  absent.  And  in- 
versely, Wherever  the  property  C is  present,  and  the  property  B 
absent,  there  the  property  A is  present. 

These  results  may  be  much  more  readily  obtained  by  the 
method  next  to  be  explained.  It  is,  however,  satisfactory  to 
possess  different  modes,  serving  for  mutual  verification,  of  ar- 
riving at  the  same  conclusion. 

4.  We  proceed  to  the  second  method. 

Proposition  II. 

If  any  equations,  1^  = 0,  V2  = 0,  Sfc.,  are  such  that  the  develop- 
ments of  their  first  members  consist  only  of  constituents  with  positive 
coefficients,  those  equations  may  be  combined  together  into  a single 
equivalent  equation  by  addition. 

For,  as  before,  let  At  represent  any  term  in  the  development 
of  the  function  Vlt  Bt  the  corresponding  term  in  the  develop- 
ment of  V2,  and  so  on.  Then  will  the  corresponding  term  in  the 
development  of  the  equation 

V1  + V2  + &c.  = 0,  (1) 

formed  by  the  addition  of  the  several  given  equations,  be 
(A  + B + &c.)  t. 

But  as  by  hypothesis  the  coefficients  A,  B,  &c.  are  none  of  them 
negative,  the  aggregate  coefficient  A + B,  &c.  in  the  derived 
equation  will  only  vanish  when  the  separate  coefficients  A,  B,  &c. 
vanish  together.  Hence  the  same  constituents  will  appear  in  the 
development  of  the  equation  (1)  as  in  the  several  equations 
V,  = 0,  V2  = 0,  &c.  of  the  original  system  taken  collectively,  and 
therefore  the  interpretation  of  the  equation  ( 1 ) will  be  cquiva- 


CHAP.  VIII.] 


OF  REDUCTION. 


121 


lent  to  the  collective  interpretations  of  the  several  equations  from 
which  it  is  derived. 

Proposition  III. 

5.  If  Vt  = 0,  V2  = 0,  Sfc.  represent  any  system  of  equations,  the 
terms  of  which  have  by  transposition  been  brought  to  the  first  side, 
then  the  combined  interpretation  of  the  system  will  be  involved  in  the 
single  equation, 

+ V22  + Sfc.  = 0, 

formed  by  adding  together  the  squares  of  the  given  equations. 

For  let  any  equation  of  the  system,  as  Fx  = 0,  produce  on  de- 
velopment an  equation 

ail  + a.2t2  + &c.  = 0, 

in  which  tx,  t2,  &c.  are  constituents,  and  au  a2 , &c.  their  corres- 
ponding coefficients.  Then  the  equation  Vd  = 0 will  produce 
on  development  an  equation 

ad  ti  + a-dt2  + &c.  = 0, 

as  may  be  proved  either  from  the  law  of  the  development  or  by 
squaring  the  function  ax  tx  + a2t2,  &c.  in  subjection  to  the  con- 
ditions 

td  = ti,  td  = t2,  t\t2  — 0, 

assigned  in  Prop.  3,  Chap.  v.  Hence  the  constituents  which 
appear  in  the  expansion  of  the  equation  Vd  = 0,  are  the  same 
with  those  which  appear  in  the  expansion  of  the  equation  Vx  = 0, 
and  they  have  positive  coefficients.  And  the  same  remark  ap- 
plies to  the  equations  V2  = 0,  &c.  Whence,  by  the  last  Propo- 
sition, the  equation 

Vd  + Vd  + &c.  = 0 

will  be  equivalent  in  interpretation  to  the  system  of  equations 
Vx  = 0,  V2=  0,  &c. 

Corollary. — Any  equation,  V=  0,  of  which  the  first  member 
already  satisfies  the  condition 

V2  = F,  or  F(1  - F)  = 0, 


122 


OF  REDUCTION. 


[CHAP.  VIII. 

does  not  need  (as  it  would  remain  unaffected  by)  the  process  of 
squaring.  Such  equations  are,  indeed,  immediately  developable 
into  a series  of  constituents,  with  coefficients  equal  to  1,  Chap.  v. 
Prop.  4. 

Proposition  IV. 

6.  Whenever  the  equations  of  a system  have  by  the  above  pro- 
cess of  squaring,  or  by  any  other  process,  been  reduced  to  a form 
such  that  all  the  constituents  exhibited  in  their  development  have 
positive  coefficients,  any  derived  equations  obtained  by  elimination 
will  possess  the  same  character,  and  may  be  combined  with  the 
other  equations  by  addition. 

Suppose  that  we  have  to  eliminate  a symbol  x from  any 
equation  V = 0,  which  is  such  that  none  of  the  constituents,  in 
the  full  development  of  its  first  member,  have  negative  coefficients. 
That  expansion  may  be  written  in  the  form 

Vxx  + F0  (1  - x)  = 0, 

Vx  and  F„  being  each  of  the  form 

axtx  -f-  a2  t2  • . h-  antn , 

in  which  tx  L . . tn  are  constituents  of  the  other  symbols,  and 
axa2. . an  in  each  case  positive  or  vanishing  quantities.  The  re- 
sult of  elimination  is 

Fx  F,  = 0; 

and  as  the  coefficients  in  Vx  and  V2  are  none  of  them  negative, 
there  can  be  no  negative  coefficients  in  the  product  Vx  V2. 
Hence  the  equation  Fa  F2  = 0 may  be  added  to  any  other  equa- 
tion, the  coefficients  of  whose  constituents  are  positive,  and  the 
resulting  equation  will  combine  the  full  significance  of  those 
from  which  it  was  obtained. 

Proposition  V. 

7.  To  deduce  from  the  previous  Propositions  a practical  rule  or 
method  for  the  reduction  of  systems  of  equations  expressing  propo- 
sitions in  Logic. 

We  have  by  the  previous  investigations  established  the  fol- 
lowing points,  viz. : 


CHAr.  VIII.] 


OF  REDUCTION. 


123 


1st.  That  any  equations  which  are  of  the  form  V = 0,  V sa- 
tisfying the  fundamental  law  of  duality  F(1  - F)  = 0,  may  be 
combined  together  by  simple  addition. 

2ndly.  That  any  other  equations  of  the  form  F=  0 may  be 
reduced,  by  the  process  of  squaring,  to  a form  in  which  the  same 
principle  of  combination  by  mere  addition  is  applicable. 

It  remains  then  only  to  determine  what  equations  in  the  ac- 
tual expression  of  propositions  belong  to  the  former,  and  what  to 
the  latter,  class. 

Now  the  general  types  of  propositions  have  been  set  forth  in 
the  conclusion  of  Chap.  iv.  The  division  of  propositions  which 
they  represent  is  as  follows  : 

1st.  Propositions,  of  which  the  subject  is  universal,  and  the 
predicate  particular. 

The  symbolical  type  (IY.  15)  is 

X = v Y, 

X and  Y satisfying  the  law  of  duality.  Eliminating  v,  we  have 

Y(1-Y)  = 0,  (1) 

and  this  will  be  found  also  to  satisfy  the  same  law.  No  further 
reduction  by  the  process  of  squaring  is  needed. 

2nd.  Propositions  of  which  both  terms,  are  universal,  and  of 
which  the  symbolical  type  is 

X = Y, 

X and  Y separately  satisfying  the  law  of  duality.  Writing  the 
equation  in  the  form  X - Y = 0,  and  squaring,  we  have 

X-2XY+  Y=  0, 

or  Y(1  - Y)  + Y(1  - X)  - 0.  (2) 

The  first  member  of  this  equation  satisfies  the  law  of  duality,  as 
is  evident  from  its  very  form. 

We  may  arrive  at  the  same  equation  in  a different  manner. 
The  equation 

X = Y 

is  equivalent  to  the  two  equations 

X = v Y,  Y=v X, 


124 


OF  REDUCTION. 


[CHAP.  VIII. 

(for  to  affirm  that  A’s  are  identical  with  Ys  is  to  affirm  both  that 
All  X’s  are  Y s,  and  that  All  Y’s  are  X’s).  Now  these  equa- 
tions give,  on  elimination  of  v, 

Ar(l-Y)  = 0,  Y(1  - X)  = 0, 

which  added,  produce  (2). 

3rd.  Propositions  of  which  both  terms  are  particular.  The 
form  of  such  propositions  is 

vX  = vY, 

but  v is  not  quite  arbitrary,  and  therefore  must  not  be  eliminated. 
For  v is  the  representative  of  some,  which,  though  it  may  include 
in  its  meaning  all , does  not  include  none.  We  must  therefore 
transpose  the  second  member  to  the  first  side,  and  square  the 
resulting  equation  according  to  the  rule. 

The  result  will  obviously  be 

vX(l  - Y)+wY(l-  X)  = 0. 

The  above  conclusions  it  may  be  convenient  to  embody  in  a 
Rule,  which  will  serve  for  constant  future  direction. 

8.  Rule. — The  equations  being  so  expressed  as  that  the  terms  X 
and  Y in  the  following  typical  forms  obey  the  law  of  duality , change 
the  equations 

X = v Y into  X (1  - Y)  = 0, 

X = Yinto  X (1  - Y)  + Y(1  - X)  = 0. 
vX  = vY  into  vX  (1  - Y)  + ?;Y(1-X)  = 0. 

Any  equation  which  is  given  in  the  form  X = 0 will  not  need  transfor- 
mation, and  any  equation  which  presents  itself  in  the  form  X = 1 
may  be  replaced  by  1 - X = 0,  as  appears  from  the  second  of  the 
above  transformations.  . 

When  the  equations  of  the  system  have  thus  been  reduced, 
any  of  them,  as  well  as  any  equations  derived  from  them  by  the 
process  of  elimination,  may  be  combined  by  addition. 

9.  Note. — It  has  been  seen  in  Chapter  iv.  that  in  literally 
translating  the  terms  of  a proposition,  without  attending  to  its 
real  meaning,  into  the  language  of  symbols,  we  may  produce 
equations  in  which  the  terms  X and  Y do  not  obey  the  law  of 
duality.  The  equation  w = st(p  + r),  given  in  (3)  Prop.  3 of 


CHAP.  VIII.] 


OF  REDUCTION. 


125 


the  chapter  referred  to,  is  of  this  kind.  Such  equations,  how- 
ever, as  it  has  been  seen,  have  a meaning.  Should  it,  for  cu- 
riosity, or  for  any  other  motive,  be  determined  to  employ  them, 
it  will  be  best  to  reduce  them  by  the  Rule  (VI.  5). 

10.  Ex.  2. — Let  us  take  the  following  Propositions  of  Ele- 
mentary Geometry : 

1st.  Similar  figures  consist  of  all  whose  corresponding  angles 
are  equal,  and  whose  corresponding  sides  are  proportional. 

2nd.  Triangles  whose  corresponding  angles  are  equal  have 
their  corresponding  sides  proportional,  and  vice  versa. 

To  represent  these  premises,  let  us  make 
s = similar. 
t - triangles. 

q = having  corresponding  angles  equal. 
r = having  corresponding  sides  proportional. 

Then  the  premises  are  expressed  by  the  following  equations  : 

s = qr,  (1) 

tq=  tr.  (2) 

Reducing  by  the  Rule,  or,  which  amounts  to  the  same  thing, 
bringing  the  terms  of  these  equations  to  the  first  side,  squaring 
each  equation,  and  then  adding,  we  have 

s + qr  - 2 qrs  + tq  + tr  - 2tqr  = 0.  (3) 

Let  it  be  required  to  deduce  a description  of  dissimilar  figures 
formed  out  of  the  elements  expressed  by  the  terms,  triangles , 
having  corresponding  angles  equal,  having  corresponding  sides 
proportional. 

We  have  from  (3), 

tq  -v  qr  + rt  - 2tqr 
S " ~ 2jr-  1 ’ 

(4) 

2 qr  - 1 v ' 

And  fully  developing  the  second  member,  we  find 
1 - s = 0 tqr  + 2 tq  (1  - r)  + 2tr(l  - q)  + t (1  - q)  (1  - r) 

+ 0(1  - t)qr  + ( 1 - t)  q (l  - r)  + (1  -<)r(l  - q) 

+ (1 -0(1 -?)(1  ->•)•  (5) 


126 


OF  REDUCTION. 


[CHAP.  VIII. 


In  the  above  development  two  of  the  terms  have  the  coefficient 
2,  these  must  be  equated  to  0 by  the  Rule,  then  those  terms 
whose  coefficients  are  0 being  rejected,  we  have 

1 - ^ =t(\  -q)  (1  -r)  + (1  -t)q  (1  - r)  + (1  - t)r  (1  - q ) 

+ (1-00 -?)(!-»■);  (6) 

tq  ( 1 - r)  = 0 ; (7  ) 

/r(l-9)  = 0;  (8) 

the  direct  interpretation  of  which  is 

1st.  Dissimilar  figures  consist  of  all  triangles  which  have  not  their 
corresponding  angles  equal  and  sides  proportional . and  of  all  figures 
not  being  triangles  which  have  either  their  angles  equal , and  sides  not 
proportional.,  or  their  corresponding  sides  proportional , and  angles 
not  equal , or  neither  their  corresponding  angles  equal  nor  corres- 
ponding sides  proportional. 

2nd.  There  are  no  triangles  whose  corresponding  angles  are  equal, 
and  sides  not  proportionals 

3rd.  There  are  no  triangles  whose  corresponding  sides  are  pro- 
portional and  angles  not  equal. 

1 1 . Such  are  the  immediate  interpretations  of  the  final  equa- 
tion. It  is  seen,  in  accordance  with  the  general  theory,  that  in 
deducing  a description  of  a particular  class  of  objects,  viz.,  dis- 
similar figures,  in  terms  of  certain  other  elements  of  the  original 
premises,  we  obtain  also  the  independent  relations  which  exist 
among  those  elements  in  virtue  of  the  same  premises.  And  that 
this  is  not  superfluous  information,  even  as  respects  the  imme- 
diate object  of  inquiry,  may  easily  be  shown.  For  example,  the 
independent  relations  may  always  be  made  use  of  to  reduce,  if  it 
be  thought  desirable,  to  a briefer  form,  the  expression  of  that  re- 
lation which  is  directly  sought.  Thus  if  we  write  (7)  in  the 
form 

0 = tq  ( 1 - r), 

and  add  it  to  (6),  we  get,  since 

t (1  - q)  (1  - r)  + tqil  - r)  = t(l  - r), 

1 - s = £(1  - r)  + (1  - t)q  (l  - r)  + (\  - t)r  (l  - q) 

+ (1  - t)  (1  - q)  (1  - r), 


OF  REDUCTION. 


127 


CHAP.  VIII.] 

which,  on  interpretation,  would  give  for  the  first  term  of  the  de- 
scription of  dissimilar. figures,  <£  Triangles  whose  corresponding 
sides  are  not  proportional,”  instead  of  the  fuller  description  origi- 
nally obtained.  A regard  to  convenience  must  always  determine 
the  propriety  of  such  reduction. 

12.  A reduction  which  is  always  advantageous  (VII.  15)  con- 
sists in  collecting  the  terms  of  the  immediate  description  sought, 
as  of  the  second  member  of  (5)  or  (6),  into  as  few  groups  as 
possible.  Thus  the  third  and  fourth  terms  of  the  second  mem- 
ber of  (6)  produce  by  addition  the  single  term  (1  - t)  (1  - q ). 
If  this  reduction  be  combined  with  the  last,  we  have 

1 - ^ = i(l  - r)  + (1  - t)q  (1  -r)  + (1  -t)  (1  - q), 
the  interpretation  of  which  is 

Dissimilar  figures  consist  of  all  triangles  whose  corresponding 
sides  are  not  proportional , and  all  figures  not  being  triangles  which 
have  either  their  corresponding  angles  unequal , or  their  corresponding 
angles  equals  but  sides  not  proportional. 

The  fulness  of  the  general  solution  is  therefore  not  a super- 
fluity. While  it  gives  us  all  the  information  that  we  seek,  it 
provides  us  also  with  the  means  of  expressing  that  information 
in  the  mode  that  is  most  advantageous. 

13.  Another  observation,  illustrative  of  a principle  which  has 
already  been  stated,  remains  to  be  made.  Two  of  the  terms  in 
the  full  development  of  1 - s in  (5)  have  2 for  their  coefficients, 

instead  of  It  will  hereafter  be  shown  that  this  circumstance 
0 

indicates  that  the  two  premises  were  not  independent.  To  verify 
this,  let  us  resume  the  equations  of  the  premises  in  their  reduced 
forms,  viz., 

s (1  - qr ) + qr  (1  - s')  - 0, 
tq  (1  - r)  + tr  (1  - q)  = 0, 

Now  if  the  first  members  of  these  equations  have  any  common 
constituents,  they  will  appear  on  multiplying  the  equations  to- 
gether. If  we  do  this  we  obtain 


stq  (1  - r ) + str{  1 - q)  = 0. 


128 


OF  REDUCTION. 


[CHAP.  VIII. 


Whence  there  will  result 

stq  (1  - r)  - 0,  str  (1  - q)  = 0, 

these  being  equations  which  are  deducible  from  either  of  the 
primitive  ones.  Their  interpretations  are — 

Similar  triangles  which  have  their  corresponding  angles  equal 
have  their  corresponding  sides  proportional. 

Similar  triangles  which  have  their  corresponding  sides  propor- 
tional have  their  corresponding  angles  equal. 

And  these  conclusions  are  equally  deducible  from  either  pre- 
miss singly.  In  this  respect,  according  to  the  definitions  laid 
down,  the  premises  are  not  independent. 

14.  Let  us,  in  conclusion,  resume  the  problem  discussed  in 
illustration  of  the  first  method  of  this  chapter,  and  endeavour  to 
ascertain,  by  the  present  method,  what  may  be  concluded  from 
the  presence  of  the  property  C,  with  reference  to  the  properties 
A and  B. 

We  found  on  eliminating  the  symbols  v the  following  equa- 
tions, viz. : 

xy  [xvz  + wz)  = 0,  (1) 

yz  ( xw  + xw)  = 0,  (2) 

xy  = wz.  (3) 

From  these  we  are  to  eliminate  w and  determine  2.  Now  (1) 
and  (2)  already  satisfy  the  condition  F(1  - V)  = 0.  The  third 
equation  gives,  on  bringing  the  terms  to  the  first  side,  and 
squaring 

xy  (1  - wz)  + w z(l  - xy)  = 0.  (4) 

Adding  (1)  (2)  and  (4)  together,  we  have 

xy(wz  + wz)  + yz(xw  +%w)+xy  (1  - wz)  + wz(  1 - xy)  = 0. 
Eliminating  w,  we  get 

{xyz  + yzx  + xy)  { xyz  + yzx  + xyz  + z ( 1 - xy)}  = 0. 

Now,  on  multiplying  the  terms  in  the  second  factor  by  those  in 
the  first  successively,  observing  that 

xx  = 0,  yy  = 0,  22  = 0, 


CHAP.  VIII.]  OF  REDUCTION.  129 

nearly  all  disappear,  and  we  have  only  left 

xyz  + xyz  = 0 ; (5) 

whence  0 

z — 

xy  + xy 

n 0 . 0_ 

= Oxy  + - xy  + - xy  + 0 xy 


0 _ 

= - xy  + 
0 y 


furnishing  the  interpretation.  Wherever  the  property  C is  found, 
either  the  property  A or  the  property  B ivill  be  found  with  it,  but 
not  both  of  them  together. 

From  the  equation  (5)  we  may  readily  deduce  the  result  ar- 
rived at  in  the  previous  investigation  by  the  method  of  arbitrary 
constant  multipliers,  as  well  as  any  other  proposed  forms  of  the 
relation  between  x,  y,  and  z ; e.  g.  If  the  property  B is  absent, 
either  A and  C will  be  jointly  present,  or  C will  be  absent.  And 
conversely,  If  A and  C are  jointly  present,  B will  be  absent. 
The  converse  part  of  this  conclusion  is  founded  on  the  presence 
of  a term  xz  with  unity  for  its  coefficient  in  the  developed  value 
of  y. 


130 


METHODS  OF  ABBREVIATION. 


[CHAP.  IX. 


CHAPTER  IX. 

ON  CERTAIN  METHODS  OF  ABBREVIATION. 

1 • 7 | ''HOUGH  the  three  fundamental  methods  of  development, 
elimination,  and  reduction,  established  and  illustrated  in 
the  previous  chapters,  are  sufficient  for  all  the  practical  ends  of 
Logic,  yet  there  are  certain  cases  in  which  they  admit,  and  espe- 
cially the  method  of  elimination,  of  being  simplified  in  an  im- 
portant degree ; and  to  these  I wish  to  direct  attention  in  the 
present  chapter.  I shall  first  demonstrate  some  propositions  in 
which  the  principles  of  the  above  methods  of  abbreviation  are 
contained,  and  I shall  afterwards  apply  them  to  particular  ex- 
amples. 

Let  us  designate  as  class  terms  any  terms  which  satisfy  the 
fundamental  law  V (1  - V)  = 0.  Such  terms  will  individually 
be  constituents ; but,  when  occurring  together,  will  not,  as  do 
the  terms  of  a development,  necessarily  involve  the  same  symbols 
in  each.  Thus  ax  + bxy  + cyz  may  be  described  as  an  expression 
consisting  of  three  class  terms,  x,  xy,  and  yz,  multiplied  by  the 
coefficients  a,  b,  c respectively.  The  principle  applied  in  the  two 
following  Propositions,  and  which,  in  some  instances,  greatly 
abbreviates  the  process  of  elimination,  is  that  of  the  rejection  of 
superfluous  class  terms ; those  being  regarded  as  superfluous 
which  do  not  add  to  the  constituents  of  the  final  result. 

Proposition  I. 

2.  From  any  equation , F=  0,  in  which  V consists  of  a series  of 
class  terms  having  positive  coefficients,  we  are  permitted  to  reject  any 
term  lohich  contains  another  term  as  a factor,  and  to  change  every 
positive  coefficient  to  unity . 

For  the  significance  of  this  series  of  positive  terms  depends 
only  upon  the  number  and  nature  of  the  constituents  of  its  final 
expansion,  i.  e.  of  its  expansion  with  reference  to  all  the  symbols 


CHAP.  IX.]  METHODS  OP  ABBREVIATION.  131 

which  it  involves,  and  not  at  all  upon  the  actual  values  of  the 
coefficients  (VI.  5).  Now  let  x be  any  term  of  the  series,  and 
xy  any  other  term  having  x as  a factor.  The  expansion  of  x Avith 
reference  to  the  symbols  x and  y will  be 

xy  + x (l  -y), 

and  the  expansion  of  the  sum  of  the  terms  x and  xy  will  be 
2xy  + x (1  - y). 

But  by  what  has  been  said,  these  expressions  occurring  in  the 
first  member  of  an  equation,  of  which  the  second  member  is  0, 
and  of  which  all  the  coefficients  of  the  first  member  are  positive, 
are  equivalent ; since  there  must  exist  simply  the  two  constituents 
xy  and  x (1  - y)  in  the  final  expansion,  whence  will  simply  arise 
the  resulting  equations 

xy  = 0,  x (1  - y)  = 0. 

And,  therefore,  the  aggregate  of  terms  x + xy  may  be  replaced  by 
the  single  term  x. 

The  same  reasoning  applies  to  all  the  cases  contemplated  in 
the  Proposition.  Thus,  if  the  term  x is  repeated,  the  aggregate 
2x  may  be  replaced  by  x,  because  under  the  circumstances  the 
equation  x = 0 must  appear  in  the  final  reduction. 

Proposition  II. 

3.  Whenever  in  the  process  of  elimination  we  have  to  multiply 
together  two  factors,  each  consisting  solely  of  positive  terms,  satisfying 
the  fundamental  laic  of  logical  symbols,  it  is  permitted  to  reject  from 
both  factors  any  common  term , or  from  either  factor  any  term  which 
is  divisible  by  a term  in  the  other  factor  ; provided  always,  that  the 
rejected  term  be  added  to  the  product  of  the  resulting  factors. 

In  the  enunciation  of  this  Proposition,  the  word  “ divisible’1 
is  a term  of  convenience,  used  in  the  algebraic  sense,  in  which  xy 
and  x (1  - y)  are  said  to  be  divisible  by  x. 

To  render  more  clear  the  import  of  this  Proposition,  let  it  be 
supposed  that  the  factors  to  be  multiplied  together  are  x + y + z 
and  x + yw  + t.  It  is  then  asserted,  that  from  these  two  factors 
we  may  reject  the  term  x,  and  that  from  the  second  factor  we 
may  reject  the  term  yw,  provided  that  these  terms  be  transferred 


132  METHODS  OF  ABBREVIATION.  [CHAP.  IX. 

to  the  final  product.  Thus,  the  resulting  factors  being  y + z 
and  t,  if  to  their  product  yt  + zt  we  add  the  terms  x and  yw, 
we  have  • 

x + yw  + yt  + zt, 

as  an  expression  equivalent  to  the  product  of  the  given  factors 
x + y + z and  x + yw  + t ; equivalent  namely  in  the  process  of 
elimination. 

Let  us  consider,  first,  the  case  in  which  the  two  factors  have 
a common  term  x,  and  let  us  represent  the  factors  by  the  expres- 
sions x + P,  x + Q,  supposing  P in  the  one  case  and  Q in  the 
other  to  be  the  sum  of  the  positive  terms  additional  to  x. 

Now, 

(x  + P)  (x  + Q)  = x + xP  + x Q + P Q.  ( 1 ) 

But  the  process  of  elimination  consists  in  multiplying  certain 
factors  together,  and  equating  the  result  to  0.  Either  then  the 
second  member  of  the  above  equation  is  to  be  equated  to  0,  or  it 
is  a factor  of  some  expression  which  is  to  be  equated  to  0. 

If  the  former  alternative  be  taken,  then,  by  the  last  Propo- 
sition, we  are  permitted  to  reject  the  terms  xP  and  rcQ,  inasmuch 
as  they  are  positive  terms  having  another  term  a;  as  a factor. 
The  resulting  expression  is 

* + PQ, 

which  is  what  we  should  obtain  by  rejecting  a;  from  both  factors, 
and  adding  it  to  the  product  of  the  factors  which  remain. 

Taking  the  second  alternative,  the  only  mode  in  which  the 
second  member  of  (1)  can  affect  the  final  result  of  elimination 
must  depend  upon  the  number  and  nature  of  its  constituents, 
both  which  elements  are  unaffected  by  the  rejection  of  the  terms 
xP  and  xQ.  For  that  development  of  a;  includes  all  possible  con- 
stituents of  which  a;  is  a factor. 

Consider  finally  the  case  in  which  one  of  the  factors  contains 
a term,  as  xy,  divisible  by  a term,  x,  in  the  other  factor. 

Let  x + P and  xy  + Q be  the  factors.  Now 

(x  + P)  (xy  + Q)  = xy  + xQ  + xyP  + PQ. 

But  by  the  reasoning  of  the  last  Proposition,  the  term  xyP  may  be 
rejected  as  containing  another  positive  term  xy  as  a factor,  whence 
we  have 


CHAP.  IX.] 


METHODS  OF  ABBREVIATION. 


133 


xy  + xQ  + PQ 
= xy  + (x  + P)  Q. 

But  this  expresses  the  rejection  of  the  term  xy  from  the  second 
factor,  and  its  transference  to  the  final  product.  Wherefore  the 
Proposition  is  manifest. 

Proposition  III. 

4.  If  t be  any  symbol  which  is  retained  in  the  final  result  of  the 
elimination  of  any  other  symbols  from  any  system  of  equations,  the  re- 
sult of  such  elimination  may  be  expressed  in  the  form 

Et  + E (l-t)  = 0, 

iu  which  E is  formed  by  maJdny  in  the  proposed  system  t = 1,  and  eli- 
minating the  same  other  symbols  ; and  E'  by  making  in  the  proposed 
system  t - 0,  and  eliminating  the  same  other  symbols. 

For  let  (f>  (t)  = 0 represent  the  final  result  of  elimination. 
Expanding  this  equation,  we  have 

</>  (1 ) t + <p  (0)  (1  - t)  = 0. 

Now  by  whatever  process  we  deduce  the  function  <f>  (<)  from  the 
proposed  system  of  equations,  by  the  same  process  should  we  de- 
duce <p  (1),  if  in  those  equations  t were  changed  into  1;  and  by 
the  same  process  should  we  deduce  (p  (0),  if  in  the  same  equations 
t were  changed  into  0.  Whence  the  truth  of  the  proposition  is 
manifest. 

5.  Of  the  three  propositions  last  proved,  it  may  be  remarked, 
that  though  quite  unessential  to  the  strict  development  or  appli- 
cation of  the  general  theory,  they  yet  accomplish  important  ends 
of  a practical  nature.  By  Prop.  1 we  can  simplify  the  results 
of  addition  ; by  Prop.  2 we  can  simplify  those  of  multiplication  ; 
and  by  Prop.  3 we  can  break  up  any  tedious  process  of  elimi- 
nation into  two  distinct  processes,  which  will  in  general  be  of  a 
much  less  complex  character.  This  method  will  be  very  fre- 
quently adopted,  when  the  final  object  of  inquiry  is  the  determi- 
nation of  the  value  of  t,  in  terms  of  the  other  symbols  which  remain 
after  the  elimination  is  performed. 

6.  Ex.  1. — Aristotle,  in  the  Nicomachean  Ethics,  Book  ii. 
Cap.  3,  having  determined  that  actions  are  virtuous,  not  as  pos- 
sessing in  themselves  a certain  character,  but  as  implying  a cer- 


134 


METHODS  OF  ABBREVIATION. 


[CHAP.  IX. 


tain  condition  of  mind  in  him  who  performs  them,  viz.,  that  he 
perform  them  knowingly,  and  with  deliberate  preference,  and  for 
their  own  sakes,  and  upon  fixed  principles  of  conduct,  proceeds 
in  the  two  following  chapters  to  consider  the  question,  whether 
virtue  is  to  be  referred  to  the  genus  of  Passions,  or  Faculties,  or 
Habits,  together  with  some  other  connected  points.  He  grounds 
his  investigation  upon  the  following  premises,  from  which,  also, 
he  deduces  the  general  doctrine  and  definition  of  moral  virtue,  of 
which  the  remainder  of  the  treatise  forms  an  exposition. 


PREMISES. 

1.  Virtue  is  either  a passion  (irdOog),  or  a faculty  (Svva/ng), 
or  a habit  (t£,ig). 

2.  Passions  are  not  things  according  to  which  we  are  praised 
or  blamed,  or  in  which  we  exercise  deliberate  preference. 

3.  Faculties  are  not  things  according  to  which  we  are  praised 
or  blamed,  and  which  are  accompanied  by  deliberate  preference. 

4.  Virtue  is  something  according  to  which  we  are  praised 
or  blamed,  and  which  is  accompanied  by  deliberate  preference. 

5.  Whatever  art  or  science  makes  its  work  to  be  in  a good 
state  avoids  extremes,  and  keeps  the  mean  in  view  relative  to 
human  nature  (to  ptaov  . . . irpdg  y)pdg'). 

6.  Virtue  is  more  exact  and  excellent  than  any  art  or  science. 

This  is  an  argument  a fortiori.  If  science  and  true  art  shun 

defect  and  extravagance  alike,  much  more  does  virtue  pursue  the 
undeviating  line  of  moderation.  If  they  cause  their  work  to  be 
in  a good  state,  much  more  reason  have  to  we  to  say  that  Virtue 
causeth  her  peculiar  work  to  be  “in  a good  state.”  Let  the 
final  premiss  be  thus  interpreted.  Let  us  also  pretermit  all  re- 
ference to  praise  or  blame,  since  the  mention  of  these  in  the  pre- 
mises accompanies  only  the  mention  of  deliberate  preference,  and 
this  is  an  element  which  we  purpose  to  retain.  We  may  then 
assume  as  our  representative  symbols — 

v = virtue. 

p - passions. 

f = faculties. 

h = habits. 

d = things  accompanied  by  deliberate  preference. 


METHODS  OF  ABBREVIATION. 


135 


CHAP.  IX.] 

g = things  causing  their  work  to  be  in  a good  state. 
m = things  keeping  the  mean  in  view  relative  to  human 
nature. 

Using,  then,  q as  an  indefinite  class  symbol,  our  premises  will  be 
expressed  by  the  following  equations  : 

v = q {p(  1 -f)  (I  - A)  +/(1  -p)  (1  - A)  + A(1  -p)  (1  -/)}. 
p=q(l  - d), 

/“?(!-  d )• 
v = qd . 
g = qrric 
v = qg. 

And  separately  eliminating  from  these  the  symbols  q, 

» I i -p  (i  -/)  0 - *)  -/( i -p)(\-h)-h(\-P)  (i  -f) =o.  (i) 


pd  = 0. 

(2) 

fd  =0. 

(3) 

v (1  - d)  = 0. 

(4) 

g(l-m)=  0. 

(5) 

v(i  -p)  = o. 

(6) 

We  shall  first  eliminate  from  (2),  (3),  and  (4)  the  symbol  d,  and 
then  determine  v in  relation  to  p,  /,  and  A.  Now  the  addition  of 
(2),  (3),  and  (4)  gives 

(p  +/)  d + v (1  - d)  = 0. 

From  which,  eliminating  d in  the  ordinary  way,  we  find 

(P+f)v  = 0.  (7) 

Adding  this  to  (1),  and  determining  v,  we  find 

0 

,’-P+/+1-)»(1-/)(1-/0-/(1-p)(1-A)-*(1 -/)(1-P)- 
Whence  by  development, 

”-5*(i-/)  (i-W- 

The  interpretation  of  this  equation  is : Virtue  is  a habit , and  not 
a faculty  or  a passion. 


136 


METHODS  OF  ABBREVIATION. 


[CHAP.  IX. 

Next,  we  will  eliminate  /,  p,  and  g from  the  original  system 
of  equations,  and  then  determine  v in  relation  to  h,  d,  and  m. 
We  will  in  this  case  eliminate  p and  /together.  On  addition  of 
(1),  (2),  and  (3),  we  get 

v [1-P(1-/)(1-^)  "/( 1-p)  (\-h)-h(\-p)  (1-/)) 

4-  pd  +fd  = 0. 

Developing  this  with  reference  to  p and  /,  we  have 

(u  + 2 d)pf+  (vh  + d)p(l-f)  + (vh  + d)  (1  - p)f 

+ v(l-h)(l-p)  (l-/)  = 0. 

Whence  the  result  of  elimination  will  be 

(v  + 2d)  ( vh  + d)  ( vh  + d)  v (1  — /*)  = 0. 

Now  v + 2d  = v+  d+d,  which  by  Prop.  I.  is  reducible  to  v + d. 
The  product  of  this  and  the  second  factor  is 

(v  + d)  (vh  + d), 

which  by  Prop.  II.  reduces  to 

d + v (vh)  or  vh  + d. 

In  like  manner,  this  result,  multiplied  by  the  third  factor,  gives 
simply  vh  + d.  Lastly,  this  multiplied  by  the  fourth  factor, 
v (1  - h),  gives,  as  the  final  equation, 

vd(l-h)  = 0.  (8) 

It  remains  to  eliminate  g from  (5)  and  (6).  The  result  is 

v (1  - m)  - 0.  (9) 

Finally,  the  equations  (4),  (8),  and  (9)  give  on  addition 

v (1  - d)  + vd  (1  - h)  + v (1  - m)  = 0, 

from  which  we  have 

0 

V ~ 1 - d + d (1  - h)  + 1 - m 
And  the  development  of  this  result  gives 

v = hdm , 

f which  the  interpretation  is, — Virtue  is  a habit  accompanied  bg 


CHAP.  IX.] 


METHODS  OF  ABBREVIATION. 


137 


deliberate  preference , and  keeping  in  vieio  the  mean  relative  to 
human  nature. 

Properly  speaking,  this  is  not  a definition,  but  a description 
of  virtue.  It  is  all , however,  that  can  be  correctly  inferred  from 
the  premises.  Aristotle  specially  connects  with  it  the  necessity 
of  prudence,  to  determine  the  safe  and  middle  line  of  action ; and 
there  is  no  doubt  that  the  ancient  theories  of  virtue  generally 
partook  more  of  an  intellectual  character  than  those  (the  theory 
of  utility  excepted)  which  have  most  prevailed  in  modern  days. 
Virtue  was  regarded  as  consisting  in  the  right  state  and  habit  of 
the  whole  mind,  rather  than  in  the  single  supremacy  of  con- 
science or  the  moral  faculty.  And  to  some  extent  those  theories 
were  undoubtedly  right.  For  though  unqualified  obedience  to 
the  dictates  of  conscience  is  an  essential  element  of  virtuous  con- 
duct, yet  the  conformity  of  those  dictates  with  those  unchanging 
principles  of  rectitude  (aidma  Socaia)  which  are  founded  in,  or 
which  rather  are  themselves  the  foundation  of  the  constitution  of 
things,  is  another  element.  And  generally  this  conformity,  in 
any  high  degree  at  least,  is  inconsistent  with  a state  of  ignorance 
and  mental  hebetude.  Reverting  to  the  particular  theory  of 
Aristotle,  it  will  probably  appear  to  most  that  it  is  of  too  ne- 
gative a character,  and  that  the  shunning  of  extremes  does  not 
afford  a sufficient  scope  for  the  expenditure  of  the  nobler  energies 
of  our  being.  Aristotle  seems  to  have  been  imperfectly  conscious 
of  this  defect  of  his  system,  when  in  the  opening  of  his  seventh 
book  he  spoke  of  an  “ heroic  virtue”*  rising  above  the  measure 
of  human  nature. 

7.  I have  already  remarked  (VIII.  1)  that  the  theory  of  sin- 
gle equations  or  propositions  comprehends  questions  which  can- 
not be  fully  answered,  .except  in  connexion  with  the  theory  of 
systems  of  equations.  This  remark  is  exemplified  when  it  is 
proposed  to  determine  from  a given  single  equation  the  relation, 
not  of  some  single  elementary  class,  but  of  some  compound  class, 
involving  in  its  expression  more  than  one  element,  in  terms  of 
the  remaining  elements.  The  following  particular  example,  and 
the  succeeding  general  problem,  are  of  this  nature. 


T>/v  inrip  •jjuag  aptrir/v  ripmKtjv  riva  icai  Qtiav. — Nic.  Etii.  Book  vii. 


138 


METHODS  OF  ABBREVIATION.  [CHAP.  IX. 


Ex.  2. — Let  us  resume  the  symbolical  expression  of  the  defi- 
nition of  wealth  employed  in  Chap,  vii.,  viz., 

w = st  [p  + r(l  - p)), 

wherein,  as  before, 


w - wealth, 

$ = things  limited  in  supply, 
t - things  transferable, 
p = things  productive  of  pleasure, 
r = things  preventive  of  pain ; 


and  suppose  it  required  to  determine  hence  the  relation  of  things 
transferable  and  productive  of  pleasure,  to  the  other  elements  of 
the  definition,  viz.,  wealth,  things  limited  in  supply,  and  things 
preventive  of  pain. 

The  expression  for  things  transferable  and  productive  of  plea- 
sure is  tp.  Let  us  represent  this  by  a new  symbol  y.  We  have 
then  the  equations 

w = st  [p  + r (1  - /?)}, 

V = 

from  which,  if  we  eliminate  t and  p,  we  may  determine  y as  a 
function  of  w,  s,  and  r.  The  result  interpreted  will  give  the  re- 
lation sought. 

Bringing  the  terms  of  these  equations  to  the  first  side,  we 
have 

w - stp  - str  (1  - p)  = 0.  , . 

y -tp  = o.  ^ ' 

And  adding  the  squares  of  these  equations  together, 
w + stp  + str  (1  - p ) - 2 wstp  - 2 icstr  (1  -p)  + y + tp  - 2ytp  = 0.  (4) 

Developing  the  first  member  with  respect  to  t and  p,  in  order  to 
eliminate  those  symbols,  we  have 


(w  + s - 2 ws  + 1 - y)tp  + (w  + sr  - 2 wsr  + y)  t (1  -p) 

+ (w  + y)  (1  - t)p  + (to  + y)  (1  - t)  (1  -p);  (5) 

and  the  result  of  the  elimination  of  t and  p will  be  obtained  by 
equating  to  0 the  product  of  the  four  coefficients  of 

tp,  t(  1 -p),  (1  - t)p , and  (1  - t)  (1  - p)- 


CHAP.  IX.]  METHODS  OF  ABBREVIATION.  139 

Or,  by  Prop.  3,  the  result  of  the  elimination  of  t and  p from  the 
above  equation  will  be  of  the  form 

Ey  + E' (l  -y), 

wherein  E is  the  result  obtained  by  changing  in  the  given  equa- 
tion y into  1 , and  then  eliminating  t and  p ; and  E'  the  result 
obtained  by  changing  in  the  same  equation  y into  0,  and  then 
eliminating  t and  p.  And  the  mode  in  each  case  of  eliminating  t 
and  p is  to  multiply  together  the  coefficients  of  the  four  con- 
stituents tp , t (1  - p) , &c. 

If  we  make  y = 1,  the  coefficients  become — 

1st.  iv  (1  - s)  + s (1  - w ). 

2nd.  1 + to  (1  - sr)  + s (1  - w)  r,  equivalent  to  1 by  Prop.  I. 
3rd  and  4th.  1 + iv,  equivalent  to  1 by  Prop.  I. 

Hence  the  value  of  E will  be 

w (1  - s)  + s (l  - w). 

Again,  in  (5)  making  y = 0,  we  have  for  the  coefficients — 
1st.  1 + w (1  - s)  + s (1  - w),  equivalent  to  1. 

2nd.  w (1  - sr)  + sr  (1  - w). 

3rd  and  4th.  w. 

The  product  of  these  coefficients  gives 
E'  = w (1  - sr). 

The  equation  from  which  y is  to  be  determined,  therefore,  is 
[w  (1  - s)  + s (1  - w))  y + iv  (1  - sr)  (1  - y)  = 0, 
tv  ( 1 - sr) 

' ' y xv  (1  - sr)  - w (1  - s)  - s (1  - xv)  5 
and  expanding  the  second  member, 

0 11 

y = - icsr  + ws  (1  - ?’)  + - w (1  - s)  r + -w(l-  s)  (1  - r) 

+ 0 (1  - w)  sr  + 0 (1  - w)  s (1  - r)  + ^ (1  - w)  (1  - s)  r 

+ 1 (i  - w)  (!  - s)  (x  - r) ; 

whence  reducing 


140 


METHODS  OF  ABBREVIATION. 


[CHAP.  IX. 


y = ws  (1  - r)  + wsr  + jj  (1  - w)  (1  - s),  (6) 

with  w (1  - s)  = 0.  (7) 

The  interpretation  of  which  is — 

1st.  Things  transferable  and  productive  of  pleasure  consist  of 
all  wealth  (limited  in  supply  and ) not  preventive  of  pain , an  inde- 
finite amount  of  wealth  ( limited  in  supply  and)  preventive  of  pain, 
and  an  indefinite  amount  of  what  is  not  wealth  and  not  limited  in 
supply. 

2nd.  All  wealth  is  limited  in  supply. 

I have  in  the  above  solution  written  in  parentheses  that  part 
of  the  full  description  which  is  implied  by  the  accompanying  in- 
dependent relation  (7). 

8.  The  following  problem  is  of  a more  general  nature,  and 
will  furnish  an  easy  practical  rule  for  problems  such  as  the  last. 

General  Problem. 

Given  any  equation  connecting  the  symbols  x,  y . .w,  z . . 

Required  to  determine  the  logical  expression  of  .any  class  ex- 
pressed in  any  way  by  the  symbols  x,  y . . in  terms  of  the  remaining 
symbols , w,  z,  &c. 

Let  us  confine  ourselves  to  the  case  in  which  there  are  but 
two  symbols,  x,  y,  and  two  symbols,  w,  z,  a case  sufficient  to  de- 
termine the  general  Rule. 

Let  V=  0 be  the  given  equation,  and  let  <p  ( x , y)  represent 
the  class  whose  expression  is  to  be  determined. 

Assume  t = <p  (x,  y),  then,  from  the  above  two  equations,  x 
and  y are  to  be  eliminated. 

Now  the  equation  V=  0 may  be  expanded  in  the  form 

Axy  + Bx  (1  - y)  + C(1  - x)  y + D (1  - x)  (1  - y)  = 0,  (1) 
A,  B,  C,  and  D being  functions  of  the  symbols  w and  z. 

Again,  as  cp  (x,  y)  represents  a class  or  collection  of  things,  it 
must  consist  of  a constituent,  or  series  of  constituents,  whose  co- 
efficients are  1. 


CHAP.  IX.]  METHODS  OF  ABBREVIATION.  141 

Wherefore  if  the  f ull  development  of  $ (x,  y)  be  represented 
in  the  form 

axy  + bx  (1  - y)  + c (1  - x)  y + d (1  - x)  (1  - y), 

the  coefficients  a , b,  c,  d must  each  be  1 or  0. 

Now  reducing  the  equation  t = <p(x,  y)  by  transposition  and 
squaring,  to  the  form 

t {l  - cj>  (x,  y)}  +<j>(x,y)(l-t)  = 0; 

and  expanding  with  reference  to  x and  y,  we  get 

{t(\  - a)  + a (\  - ())  xy  + {£(l-&)  + 6(l  -£)}  a;  ( 1 - y) 

+ {*  0 ~0  + c(l  - 0)  (i  - x)y 

+ [t{\-  d)  + d(\ -t))  (1  - x)  (1  - y)  = 0; 

whence,  adding  this  to  (1),  we  have 

{A  + t (1  - a)  + a (1  - 1) } xy 

4-  {J3  + £(l-6)  + 6(l-^)j  a-  (1  -?/)  + &c.  = 0. 

Let  the  result  of  the  elimination  of  x and  y be  of  the  form 

Et  + E'(l  - t)  = 0, 

then  E will,  by  what  has  been  said,  be  the  reduced  product  of 
what  the  coefficients  of  the  above  expansion  become  when  t = 1 , 
and  E the  product  of  the  same  factors  similarly  reduced  by  the 
condition  t = 0. 

Hence  E will  be  the  reduced  product 

(A  + 1 - a)  (B+  1 - b)  (C  + 1 -c)  (Z>+  1 - d). 

Considering  any  factor  of  this  expression,  as  A + 1 - a,  we  see 
that  when  a = 1 it  becomes  A,  and  when  a = 0 it  becomes  1 + A, 
which  reduces  by  Prop.  I.  to  1.  Hence  we  may  infer  that  E will 
be  the  product  of  the  coefficients  of  those  constituents  in  the  de- 
velopment of  V whose  coefficients  in  the  development  of  <p  ( x , y) 
are  1. 

Moreover  E'  will  be  the  reduced  product 

(A  + a)  (B  + b)(C+c)(D  + d). 

Considering  any  one  of  these  factors,  as  A + a,  we  see  that  this 
becomes  A when  a = 0,  and  reduces  to  1 when  a = 1 ; and  so  on 
for  the  others.  Hence  E will  be  the  product  of  the  coefficients 


142 


METHODS  OF  ABBREVIATION. 


[C 


HAP.  IX. 


of  those  constituents  in  the  development  of  y,  whose  coefficients 
in  the  development  <£  ( x , y)  are  0.  Viewing  these  cases  together, 
we  may  establish  the  following  Rule : 

9.  To  deduce  from  a logical  equation  the  relation  of  any  class 
expressed  by  a given  combination  of  the  symbols  x,  y,  Sfc,  to  the 
classes  represented  by  any  other  symbols  involved  in  the  given 
equation . 

Rule. — Expand  the  given  equation  with  reference  to  the  sym- 
bols x,  y.  Then  form  the  equation 

Et  + E'{\  - t)  = 0, 

in  ivhich  E is  the  product  of  the  coefficients  of  all  those  constituents 
in  the  above  development , whose  coefficients  in  the  expression  of  the 
given  class  are  1,  and  E the  product  of  the  coefficients  of  those  con- 
stituents of  the  development  whose  coefficients  in  the  expression  of  the 
given  class  are  0.  The  value  of  t deduced  from  the  above  equation 
by  solution  and  interpretation  will  be  the  expression  required. 

Note. — Although  in  the  demonstration  of  this  Ride  V is  sup- 
posed to  consist  solely  of  positive  terms , it  may  easily  be  shown  that 
this  condition  is  unnecessary,  and  the  Rule  general,  and  that  no  pre- 
paration of  the  given  equation  is  really  required. 

10.  Ex.  3. — The  same  definition  of  wealth  being  given  as  in 
Example  2,  required  an  expression  for  things  transferable,  but  not 
productive  of  pleasure,  £ (1  - p),  in  terms  of  the  other  elements 
represented  by  iv,  s,  and  r. 

The  equation 

to  - sip  - sir  ( 1 - p)  = 0, 
gives,  when  squared, 

w + stp  + str  (1  - p)  - 2wstp  - 2 ivstr  (1  - p)  = 0 ; 

and  developing  the  first  member  with  respect  to  t and  p, 

(■ w + s - 2 ws)  tp  + ( iv  + sr  - 2 wsr)  t (1  - p)  + w (1  - 1)  p 

+ w (1  - t)  (1  - p)  = 0. 

The  coefficients  of  which  it  is  best  to  exhibit  as  in  the  following 
equation ; 

{w>(l-s)  + s(1^0)}(p  + {?y(l-sr)  + sr(l-w))  t(\-p)  + w (f-t)p 

+ jo  (1  - t)  (1  - p)  = 0. 


CHAP.  IX.] 


METHODS  OF  ABBREVIATION. 


143 


Let  the  function  t (1  - p)  to  be  determined,  be  represented  by  z ; 
then  the  full  development  of  z in  respect  of  t and  p is 

z = 0 tp  + t (1  - p)  + 0 (1  - t)  p + 0 (1  - t)  (1  - p). 

Hence,  by  the  last  problem,  we  have 
Ez  + E'  (1  - z)  = 0 ; 

where  E = to  (1  — sr ) + sr  (1  - to)  ; 

E'  = { to  (1  - s)  + s (1  - to) } xtoxto  = to(l-s); 

.’.  {iv  (1  - sr)  + sr  (1  - to)}  z + to  (1  - s)  (1  - z)  = 0. 

Hence, 

to  (1  - s) 

Z = 4- 

2wsr  — ws  - sr 

= ^ icsr  + 0 ivs  (1  - r)  + ^ w (1  - s)  r + ^ to  (1  - s)  (1  - r), 

+ 0 (1  - w)  sr  + jj  (1  - to)  s (1  - r)  + ^ (1  - to)  (1  - s)  r 

+ (1  - w)  (1  - s)  (1  - r). 

0r’  ' 2 U wsr  + (!  ~ «’)  * U - r)  + ^ (1  - to)  (1  - s), 

with  to  (1  - s)  = 0. 

Hence,  Things  transferable  and  not  productive  of  pleasure  are 
either  wealth  ( limited  in  supply  and  preventive  of  pain) ; or  things 
which  are  not  wealth , but  limited  in  supply  and  not  preventive  of 
pain;  or  things  which  are  not  wealth,  and  are  unlimited  in  supply. 

The  following  results,  deduced  in  a similar  manner,  will  be 
easily  verified : 

Things  limited  in  supply  and  productive  of  pleasure  which  are 
not  wealth, — are  intransfer  able. 

Wealth  that  is  not  productive  of  pleasure  is  transferable,  limited 
in  supply,  and  preventive  of  pain. 

Things  limited  in  supply  which  are  either  wealth,  or  are  pro- 
ductive of  pleasure,  but  not  both, — are  either  transferable  and  pre- 
ventive of  pain , or  intransferable. 

11.  F rom  the  domain  of  natural  history  a large  number  of 
curious  examples  might  be  selected.  I do  not,  however,  con- 


144 


METHODS  OF  ABBREVIATION. 


[CHAP.  IX. 

ceive  that  such  applications  would  possess  any  independent  va- 
lue. They  would,  for  instance,  throw  no  light  upon  the  true 
principles  of  classification  in  the  science  of  zoology.  For  the 
discovery  of  these,  some  basis  of  positive  knowledge  is  requisite* — 
some  acquaintance  with  organic  structure,  with  teleological  adap- 
tation ; and  this  is  a species  of  knowledge  which  can  only  be  de- 
rived from  the  use  of  external  means  of  observation  and  analysis. 
Taking,  however,  any  collection  of  propositions  in  natural  his- 
tory, a great  number  of  logical  problems  present  themselves, 
without  regard  to  the  system  of  classification  adopted.  Perhaps 
in  forming  such  examples,  it  is  better  to  avoid,  as  superfluous, 
the  mention  of  that  property  of  a class  or  species  which  is  im- 
mediately suggested  by  its  name,  e.  g.  the  ring-structure  in  the 
annelida,  a class  of  animals  including  the  earth-worm  and  the 
leech. 

Ex.  4. — 1*.  The  annelida  are  soft-bodied,  and  either  naked  or 
enclosed  in  a tube. 

2.  The  annelida  consist  of  all  invertebrate  animals  having 
red  blood  in  a double  system  of  circulating  vessels. 

Assume  a = annelida ; 

s = soft-bodied  animals  ; 
n = naked ; 

t = enclosed  in  a tube ; 
i = invertebrate ; 
r = having  red  blood,  <£c. 

Then  the  propositions  given  will  be  expressed  by  the  equations 
a = vs  [n  (1  - t)  + t (1  - ft)}  ; (1) 

a = ir;  (2) 

to  which  we  may  add  the  implied  condition, 

nt  = 0.  (3) 

On  eliminating  v,  and  reducing  the  system  to  a single  equation, 
we  have 

a { 1 - sn  (1  - t)  - st  ( 1 - n) } + a (1  - ir)  + ir  (1  - a)  + nt  - 0.  (4) 

Suppose  that  we  wish  to  obtain  the  relation  in  which  soft- 
bodied  animals  enclosed  in  tubes  are  placed  (by  virtue  of  the 


CHAP.  XX.] 


METHODS  OF  ABBREVIATION. 


145 


premises)  with  respect  to  the  following  elements,  viz.,  the  pos- 
session of  red  blood,  of  an  external  covering,  and  of  a vertebral 
column. 

We  must  first  eliminate  a.  The  result  is 

ir  { 1 - sn  (1  - t)  - st  (1  - n)\  + nt  = 0. 

Then  (IX.  9)  developing  with  respect  to  s and  t,  and  reducing 
the  first  coefficient  by  Prop.  1,  we  have 

nst  + ?>(l-n)s(l-  t ) + (ir  + »)(!-)*  + ?V(l-s)(l  -/)  = 0.  (5) 

Hence,  if  st  = w,  we  find 

nw  + ir  ( 1 - n)  x (ir  + n)xir(l-w)  = 0; 
or,  nw  + ir  (1  - n)  (1  - w)  = 0 ; 

ir  (l  - n) 

♦ W = r 

ir  ( 1 - n)  - n 

= 0 irn  + ir  (1  - n)  + 0*  (1  - r)  n + i (1  - r)  (1  - n ) 

+ 0(1-?)  rn  + jj  (1  - i)  r (1  - n)  4 0 ( 1 - i)  (1  - r)  n 

w = ir  (1  - n)  + H i (1  - r)  (1  - n)  + (1  - i)  (1  - n ). 

Hence,  soft-bodied  animals  enclosed  in  tubes  consist  of  all 
invertebrate  animals  having  red  blood  and  not  naked,  and  an  in- 
definite remainder  of  invertebrate  animals  not  having  red  blood  and 
not  naked,  and  of  vertebrate  animals  which  are  not  naked. 

And  in  an  exactly  similar  manner,  the  following  reduced  equa- 
tions, the  interpretation  of  which  is  left  to  the  reader,  have  been 
deduced  from  the  development  (5). 

s (1  - £)  = irn  + * (1  - n)  + jj  (l  - i) 

0 - *)  t = l 0 - 0 r (1  “ n)  + (1  - r)  l1  ~ n) 

(1  - «)  (1  - 0 = 5 ?'0  “ r)  + 1 (l  -0- 


14b 


METHODS  OF  ABBREVIATION. 


CHAP.  IX. 


In  none  of  the  above  examples  has  it  been  my  object  to  ex- 
hibit in  any  special  manner  the  power  of  the  method.  That, 
I conceive,  can  only  be  fully  displayed  in  connexion  with  the 
mathematical  theory  of  probabilities.  I would,  however,  suggest 
to  any  who  may  be  desirous  of  forming  a correct  opinion  upon 
this  point,  that  they  examine  by  the  rules  of  ordinary  logic  the 
following  problem,  before  inspecting  its  solution ; remembering 
at  the  same  time,  that  whatever  complexity  it  possesses  might 
be  multiplied  indefinitely,  with  no  other  effect  than  to  render  its 
solution  by  the  method  of  this  work  more  operose,  but  not  less 
certainly  attainable. 

Ex.  5.  Let  the  observation  of  a class  of  natural  productions 
be  supposed  to  have  led  to  the  following  general  results. 

1st,  That  in  whichsoever  of  these  productions  the  properties 
A and  C are  missing,  the  property  E is  found,  together  with  one 
of  the  properties  B and  D,  but  not  with  both. 

2nd,  That  wherever  the  properties  A and  D are  found  while 
E is  missing,  the  properties  B and  C will  either  both  be  found, 
or  both  be  missing. 

3rd,  That  wherever  the  property  A is  found  in  conjunction 
with  either  B or  E , or  both  of  them,  there  either  the  property 
C or  the  property  1)  will  be  found,  but  not  both  of  them.  And 
conversely,  wherever  the  property  C or  D is  found  singly,  there 
the  property  A will  be  found  in  conjunction  with  either  B or  E , 
or  both  of  them. 

Let  it  then  be  required  to  ascertain,  first,  what  in  any  parti- 
cular instance  may  be  concluded  from  the  ascertained  presence  of 
the  property  A,  with  reference  to  the  properties  B,  C,  and  B ; 
also  whether  any  relations  exist  independently  among  the  pro- 
perties B,  C,  and  D.  Secondly,  what  may  be  concluded  in  like 
manner  respecting  the  property  B,  and  the  properties  A,  C, 
and  D. 

It  will  be  observed,  that  in  each  of  the  three  data,  the  informa- 
tion,conveyed  respecting  the  properties  A,  B , C , and  D,  is  com- 
plicated with  another  element,  E , about  which  we  desire  to  say 
nothing  in  our  conclusion.  It  will  hence  be  requisite  to  eliminate 
the  symbol  representing  the  property  E from  the  system  of  equa- 
tions, by  which  the  given  propositions  will  be  expressed. 


CHAP.  IX.]  METHODS  OF  ABBREVIATION.  147 

Let  us  represent  the  property  A by  x,  B by  y,  Chj  z,  D by 
w,  E by  v.  The  data  are 

~xz  = qv  (yw  + wy);  (1) 

vxw  = q (yz  + yz)\  (2) 

xy  + xvy  = wz  + zw ; (3) 

x standing  for  1 - x,  &c.,  and  q being  an  indefinite  class  symbol. 
Eliminating  q separately  from  the  first  and  second  equations, 
and  adding  the  results  to  the  third  equation  reduced  by  (5), 
Chap/vm.,  we  get 

xz(l  - vyw  - vwy)  + vxw  {jyz  + zy)  + (xy  + xvy)  (wz  + wz) 

+ (wz  + zw)  (1  - xy  - xvy)  = 0.  (4) 

From  this  equation  v must  be  eliminated,  and  the  value  of  x 
determined  from  the  result.  For  effecting  this  object9  it  will 
be  convenient  to  employ  the  method  of  Prop.  3 of  the  present 
chapter. 

Let  then  the  result  of  elimination  be  represented  by  the 
equation 

Ex  + E'  (1  -x)  - 0. 

To  find  E make  x = 1 in  the  first  member  of  (4),  we  find 
vw  (yz  + zy)  + (y  + vy)  (wz  + wz)  + (wz  + zw)  vy. 
Eliminating  v,  we  have 

(wz  + wT)  [w  (yz  + zy)  + y (wz  + wT)  + y (wz  + zw) } ; 

which,  on  actual  multiplication,  in  accordance  with  the  conditions 
ww  = 0,  zz  = 0,  &c.,  gives 

E = wz  + ywz. 

Next,  to  find  E'make  2 = 0 in  (4),  we  have 

z (1  - vyw  - vyw)  + wz  + zw . 

whence,  eliminating  v , and  reducing  the  result  by  Propositions 
1 and  2,  we  find 

E = wz  + zw  +ywzm, 
and,  therefore,  finally  we  have 

(wz  + ywz)  x + (wz  + zw  + ywz)  2 = 0; 


from  which 


(5) 


148 


METHODS  OF  ABBREVIATION. 


[CHAP.  IX. 


wz  + zio  + y w z 
wz  + zw  + ywz  -wz  - ywz  ’ 
wherefore,  by  development, 

x = 0 yzw  + yzw  + yzw  + 0 yzw 
+ 0 yzw  + yzw  + yzw  + yzl", 
or,  collecting  the  terms  in  vertical  columns, 

x = zw  + zw  + yzw ; (6) 

the  interpretation  of  which  is^ — 

In  whatever  submances  the  property  A is  found , there  will  also 
be  found  either  the  property  C or  the  property  D,  but  not  both,  or 
else  the  properties  B,  C,  and  D,  will  all  be  wanting.  And  con- 
versely, where  either  the  property  C or  the  property  D is  found 
singly,  or  the  properties  B,  C,  and  D,  are  together  missing,  there 
the  property  A will  be  found. 

It  also  appears  that  there  is  no  independent  relation  among 
the  properties  B,  C,  and  D. 

Secondly,  we  are  to  find  y.  Now  developing  (5)  with  respect 
to  that  symbol, 

( xivz  + x wz  + xwz  + xzw)  y + ( xwz  + xwz  + xzw  + xzw)  y = 0 ; 
whence,  proceeding  as  before, 


y = xwz  + ^ (xwz  + xwz  + xzw), 

(V) 

xziv  = 0 ; 

(8) 

xzw  - 0 ; 

0) 

xzw  - 0 ; 

(10) 

From  (10)  reduced  by  solution  to  the  form 

0 

xz  = - w ; 

we  have  the  independent  relation, — If  the  property  A is  absent 
and  C present,  D is  present.  Again,  by  addition  and  solution  (8) 
and  (9)  give 

__  0 _ 
xz  + xz  = - w. 

Whence  wc  have  for  the  general  solution  and  the  remaining  in- 
dependent relation  : 


METHODS  OF  ABBREVIATION. 


149 


CHAP.  IX.] 

1st.  If  the  property  B be  present  in  one  of  the  productions,  either 
tht  properties  A,  C,  and  D,  are  all  absent,  or  some  one  alone  of  them 
is  absent.  And  conversely,  if  they  are  all  absent  it  may  be  con- 
cluded that  the  property  A is  present  (7). 

2nd.  If  A and  C are  both  present  or  both  absent,  D will  be  ab- 
sent, quite  independently  of  the  presence  or  absence  of  B (8)  and  (9). 

I have  not  attempted  to  verify  these  conclusions. 


150 


CONDITIONS  OF  A PERFECT  METHOD.  [CHAP.  X. 


CHAPTER  X. 

OF  THE  CONDITIONS  OF  A PERFECT  METHOD. 

1.  HPHE  subject  of  Primary  Propositions  has  been  discussed  at 
length,  and  we  are  about  to  enter  upon  the  consideration 
of  Secondary  Propositions.  The  interval  of  transition  between 
these  two  great  divisions  of  the  science  of  Logic  may  afford  a fit 
occasion  for  us  to  pause,  and  while  reviewing  some  of  the  past 
steps  of  our  progress,  to  inquire  what  it  is  that  in  a subject  like 
that  with  which  we  have  been  occupied  constitutes  perfection  of 
method.  I do  not  here  speak  of  that  perfection  only  which  con- 
sists in  power,  but  of  that  also  which  is  founded  in  the  conception 
of  what  is  fit  and  beautiful.  It  is  probable  that  a careful  analysis 
of  this  question  would  conduct  us  to  some  such  conclusion  as  the 
following,  viz.,  that  a perfect  method  should  not  only  be  an  effi- 
cient one,  as  respects  the  accomplishment  of  the  objects  for  which 
it  is  designed,  but  should  in  all  its  parts  and  processes  manifest 
a certain  unity  and  harmony.  This  conception  would  be  most, 
fully  realized  if  even  the  very  forms  of  the  method  were  sugges- 
tive of  the  fundamental  principles,  and  if  possible  of  the  one  fun- 
damental principle,  upon  which  they  are  founded.  In  applying 
these  considerations  to  the  science  of  Reasoning,  it  may  be  well 
to  extend  our  view  beyond  the  mere  analytical  processes,  and  to 
inquire  what  is  best  as  respects  not  only  the  mode  or  form  of 
deduction,  but  also  the  system  of  data  or  premises  from  which 
the  deduction  is  to  be  made. 

2.  As  respects  mere  power,  there  is  no  doubt  that  the  first 
of  the  methods  developed  in  Chapter  vm.  is,  within  its  proper 
sphere,  a perfect  one.  The  introduction  of  arbitrary  constants 
makes  us  independent  of  the  forms  of  the  premises,  as  well  as  of 
any  conditions  among  the  equations  by  which  they  are  repre- 
sented. But  it  seems  to  introduce  a foreign  element,  and  while 
it  is  a more  laborious,  it  is  also  a less  elegant  form  of  solution 
than  the  second  method  of  reduction  demonstrated  in  the  same 


CHAI\  X.]  CONDITIONS  Of  A PERFECT  METHOD.  151 

chapter.  There  are,  however,  conditions  under  which  the  latter 
method  assumes  a more  perfect  form  than  it  otherwise  bears.  To 
make  the  one  fundamental  condition  expressed  by  the  equation 

* (1  - a?)  = 0, 

the  universal  type  of  form,  would  give  a unity  of  character  to 
both  processes  and  results,  which  would  not  else  be  attainable. 
Were  brevity  or  convenience  the  only  valuable  quality  of  a me- 
thod, no  advantage  would  flow  from  the  adoption  of  such  a prin- 
ciple. For  to  impose  upon  every  step  of  a solution  the  character 
above  described,  would  involve  in  some  instances  no  slight  la- 
bour of  preliminary  reduction.  But  it  is  still  interesting  to  know 
that  this  can  be  done,  and  it  is  even  of  some  importance  to  be 
acquainted  with  the  conditions  under  which  such  a form  of  solu- 
tion would  spontaneously  present  itself.  Some  of  these  points 
will  be  considered  in  the  present  chapter. 

Proposition  I. 

3.  To  reduce  any  equation  among  logical  symbols  to  the  form 
F = 0,  in  ivliich  V satisfies  the  law  of  duality , 

F(1  - F)  = 0. 

It  is  shown  in  Chap.  v.  Prop.  4,  that  the  above  condition  is 
satisfied  whenever  V is  the  sum  of  a series  of  constituents.  And 
it  is  evident  from  Prop.  2,  Chap.  vi.  that  all  equations  are  equi- 
valent which,  when  reduced  by  transposition  to  the  form  F = 0, 
produce,  by  development  of  the  first  member,  the  same  series  of 
constituents  with  coefficients  which  do  not  vanish ; the  particular 
numerical  values  of  those  coefficients  being  immaterial. 

Hence  the  object  of  this  Proposition  may  always  be  accom- 
plished by  bringing  all  the  terms  of  an  equation  to  the  first  side, 
fully  expanding  that  member,  and  changing  in  the  result  all  the  co- 
efficients which  do  not  vanish  into  unity,  except  such  as  have  already 
that  value. 

But  as  the  development  of  functions  containing  many  sym- 
bols conducts  us  to  expressions  inconvenient  from  their  great 


152 


CONDITIONS  OF  A PERFECT  METHOD.  [CHAP.  X. 

length,  it  is  desirable  to  show  how,  in  the  only  cases  which  do 
practically  offer  themselves  to  our  notice,  this  source  of  com- 
plexity may  be  avoided. 

The  great  primary  forms  of  equations  have  already  been  dis- 
cussed in  Chapter  vm.  They  are — 

X = vY, 

x = y, 

vX  = vY. 

Whenever  the  conditions  X (1  - X)  = 0,  Y (1  - Y)  = 0,  are 
satisfied,  we  have  seen  that  the  two  first  of  the  above  equations 
conduct  us  to  the  forms 

X(1-Y)  = 0,  (1) 

X(l-  Y)  + Y (1  - X)  = 0 ; (2) 

and  under  the  same  circumstances  it  may  be  shown  that  the  last 
of  them  gives 

t>  [X(l-  Y)+  Y(l-  X)}  =0;  (3) 

all  which  results  obviously  satisfy,  in  their  first  members,  the 
condition 

F(1  - V)  = 0. 

Now  as  the  above  are  the  forms  and  conditions  under  which  the 
equations  of  a logical  system  properly  expressed  do  actually  pre- 
sent themselves,  it  is  always  possible  to  reduce  them  by  the 
above  method  into  subjection  to  the  law  required.  Though, 
however,  the  separate  equations  may  thus  satisfy  the  law,  their 
equivalent  sum  (VIII.  4)  may  not  do  so,  and  it  remains  to 
show  how  upon  it  also  the  requisite  condition  may  be  imposed. 

Let  us  then  represent  the  equation  formed  by  adding  the 
several  reduced  equations  of  the  system  together,  in  the  form 

v + v + v",  &c.  = 0,  (4) 

this  equation  being  singly  equivalent  tc  the  system  from  which 
it  was  obtained.  We  suppose  v,  v',  v",  &c.  to  be  class  terms 
(IX.  1 ) satisfying  the  conditions 

v (1  - v)  = 0,  v (1  - v')  = 0,  &c. 

Now  the  full  interpretation  of  (4)  would  be  found  by  deve- 


CHAP.  X.]  CONDITIONS  OF  A PERFECT  METHOD. 


153 


loping  the  first  member  with  respect  to  all  the  elementary  symbols 
x,  y,  &c.  which  it  contains,  and  equating  to  0 all  the  constituents 
whose  coefficients  do  not  vanish ; in  other  words,  all  the  consti- 
tuents which  are  found  in  either  v,  ?/,  v",  &c.  But  those  consti- 
tuents consist  of — 1st,  such  as  are  found  in  v ; 2nd,  such  as  are 
not  found  in  v,  but  are  found  in  v' ; 3rd,  such  as  are  neither  found 
in  v nor  v,  but  are  found  in  v",  and  so  on.  Hence  they  will  be 
such  as  are  found  in  the  expression 

v + (1  - v)  v + (1  - v)  (1  - v ) v"  + &c.,  (5) 

an  expression  in  which  no  constituents  are  repeated,  and  which 
obviously  satisfies  the  law  F(1  - V)  = 0. 

Thus  if  we  had  the  expression 

(1  - i)  + v + (1  - z)  + tzw, 

in  which  the  terms  1 - t,  1 - z are  bracketed  to  indicate  that  they 
are  to  be  taken  as  single  class  terms,  we  should,  in  accordance 
with  (5),  reduce  it  to  an  expression  satisfying  the  condition 
V (l  - V)  = 0,  by  multiplying  all  the  terms  after  the  first  by  t, 
then  all  after  the  second  by  1 - v ; lastly,  the  term  which  remains 
after  the  third  by  z ; the  result  being 

1 - t + tv  + t (1  - v)  (1  - z)  + t (1  - v)  zw.  (6) 

4.  All  logical  equations  then  are  reducible  to  the  form  V-  0, 
V satisfying  the  law  of  duality.  But  it  would  obviously  be  a 
higher  degree  of  perfection  if  equations  always  presented  them- 
selves in  such  a form,  without  preparation  of  any  kind,  and  not 
only  exhibited  this  form  in  their  original  statement,  but  retained 
it  unimpaired  after  those  additions  which  are  necessary  in  order 
to  reduce  systems  of  equations  to  single  equivalent  forms.  That 
they  do  not  spontaneously  present  this  feature  is  not  properly 
attributable  to  defect  of  method,  but  is  a consequence  of  the  fact 
that  our  premises  are  not  always  complete,  and  accurate,  and  in- 
dependent. They  are  not  complete  when  they  involve  material 
(as  distinguished  from  formal)  relations,  which  are  not  expressed. 
They  are  not  accurate  when  they  imply  relations  which  are  not 
intended.  But  setting  aside  these  points,  with  which,  in  the 
present  instance,  we  are  less  concerned,  let  it  be  considered  in 
what  sense  they  may  fail  of  being  independent. 


154 


CONDITIONS  OF  A PERFECT  METHOD.  [CHAP.  X. 

5.  A system  of  propositions  may  be  termed  independent, 
when  it  is  not  possible  to  deduce  from  any  portion  of  the  system 
a conclusion  deducible  from  any  other  portion  of  it.  Supposing 
the  equations  representing  those  propositions  all  reduced  to  the 
form 

7=0, 

then  the  above  condition  implies  that  no  constituent  which  can 
be  made  to  appear  in  the  development  of  a particular  function  7 
of  the  system,  can  be  made  to  appear  in  the  development  of  any 
other  function  V of  the  same  system.  When  this  condition  is 
not  satisfied,  the  equations  of  the  system  are  not  independent. 
This  may  happen  in  various  cases.  Let  all  the  equations  satisfy 
in  their  first  members  the  law  of  duality,  then  if  there  appears  a 
positive  term  x in  the  expansion  of  one  equation,  and  a term  xy 
in  that  of  another,  the  equations  are  not  independent,  for  the 
term  x is  further  developable  into  xy  + x ( 1 - y),  and  the  equation 

xy  = 0 

is  thus  involved  in  both  the  equations  of  the  system.  Again,  let 
a term  xy  appear  in  one  equation,  and  a term  xz ' in  another. 
Both  these  may  be  developed  so  as  to  give  the  common  consti- 
tuent xyz.  And  other  cases  may  easily  be  imagined  in  which 
premises  which  appear  at  first  sight  to  be  quite  independent  are 
not  really  so.  Whenever  equations  of  the  form  7=0  are  thus 
not  truly  independent,  though  individually  they  may  satisfy  the 
law  of  duality, 

7(1  - 7)  = 0, 

the  equivalent  equation  obtained  by  adding  them  together  will 
not  satisfy  that  condition,  unless  sufficient  reductions  by  the  me- 
thod of  the  present  chapter  have  been  performed.  When,  on 
the  other  hand,  the  equations  of  a system  both  satisfy  the  above 
law,  and  are  independent  of  each  other,  their  sum  will  also  sa- 
tisfy the  same  law.  I have  dwelt  upon  these  points  at  greater 
length  than  would  otherwise  have  been  necessary,  because  it  ap- 
pears to  me  to  be  important  to  endeavour  to  form  to  ourselves, 
and  to  keep  before  us  in  all  our  investigations,  the  pattern  of  an 
ideal  perfection, — the  object  and  the  guide  of  future  efforts.  In 


CHAP.  X.] 


CONDITIONS  OF  A PERFECT  METHOD. 


155 


the  present  class  of  inquiries  the  chief  aim  of  improvement  of  me- 
thod should  be  to  facilitate,  as  far  as  is  consistent  with  brevity, 
the  transformation  of  equations,  so  as  to  make  the  fundamental 
condition  above  adverted  to  universal. 

In  connexion  with  this  subject  the  following  Propositions  are 
deserving  of  attention. 

Proposition  II. 


If  the  first  member  of  any  equation  V = 0 satisfy  the  condition 
F(1  - V)  = 0,  and  if  the  expression  of  any  symbol  t of  that  equa- 
tion be  determined  as  a developed  function  of  the  other  symbols , the 


coefficients  of  the  expansion  can  only  assume  the  forms  1,  0 


0 1 
’ 0’  0‘ 


For  if  the  equation  be  expanded  with  reference  to  t,  we  ob- 
tain as  the  result, 

Et  + E'(\-t),  (1) 

E and  E'  being  what  1 ' becomes  when  t is  successively  changed 
therein  into  1 and  0.  Hence  E and  E will  themselves  satisfy 
the  conditions 


Now  (1)  gives 


E(l-E)  = 0,  E\\  - E')  = 0. 
E' 

l~  E - E 


(2) 


the  second  member  of  which  is  to  be  expanded  as  a function  of 
the  remaining  symbols.  It  is  evident  that  the  only  numerical 
values  which  E and  E'  can  receive  in  the  calculation  of  the  co- 
efficients will  be  1 and  0.  The  following  cases  alone  can  there- 


fore  arise : 

1st. 

E'=  1, 

E=  1, 

then 

E' 

l 

E'  - E~ 

O' 

2nd. 

E = 1, 

E=  0, 

then 

E' 

1. 

E - E 

3rd. 

ii 

© 

E=  1, 

then 

E 

0. 

tq 

i 

i 

4 th. 

tq 

II 

© 

© 

II 

then 

E 

0 

1 

i 

o' 

Whence  the  truth  of  the  Proposition  is  manifest. 


156 


CONDITIONS  OF  A PERFECT  METHOD.  [CHAP.  X. 


6.  It  may  be  remarked  that  the  forms  1,  0,  and  ^ appear  in 
the  solution  of  equations  independently  of  any  reference  to  the 
condition  F(1  - F)  = 0.  But  it  is  not  so  with  the  coefficient 

The  terms  to  which  this  coefficient  is  attached  when  the  above 
condition  is  satisfied  may  receive  any  other  value  except  the 

three  values  1,  0,  and  whfen  that  condition  is  not  satisfied.  It 


is  permitted,  and  it  would  conduce  to  uniformity,  to  change  any 
coefficient  of  a development  not  presenting  itself  in  any  of  the 

four  forms  referred  to  in  this  Proposition  into  regarding  this 


as  the  symbol  proper  to  indicate  that  the  coefficient  to  which  it  is 
attached  should  be  equated  to  0.  This  course  I shall  frequently 
adopt. 

Proposition  III. 

7.  The  result  of  the  elimination  of  any  symbols  x,  y,  8fc.from 
an  equation  V = 0,  of  which  the  first  member  identically  satisfies 
the  law  of  duality , 

V(  1 - F)  = 0, 

may  be  obtained  by  developing  the  given  equation  with  reference  to 
the  other  symbols , and  equating  to  0 the  sum  of  those  constituents 
whose  coefficients  in  the  expansion  are  equal  to  unity. 


Suppose  that  the  given  equation  F = 0 involves  but  three 
symbols,  x,  y,  and  t,  of  which  x and  y are  to  be  eliminated.  Let 
the  development  of  the  equation,  with  respect  to  t,  be 

At  + B(\-t)  = 0,  (1) 

A and  B being  free  from  the  symbol  t. 

By  Chap.  ix.  Prop.  3,  the  result  of  the  elimination  of*  and  y 
from  the  given  equation  will  be  of  the  form 


Et+  E(\  -t)  = 0,  (2) 

in  which  E is  the  result  obtained  by  eliminating  the  symbols  x 
and  y from  the  equation  A = 0,  E'  the  result  obtained  by  elimi- 
nating from  the  equation  B *=  0. 


157 


CHAP.  X.]  CONDITIONS  OF  A PERFECT  METHOD. 

Now  A and  B must  satisfy  the  condition 

J(1-A)  = 0,  -B  (1  - JB)  = 0. 

Hence  A (confining  ourselves  for  the  present  to  this  coefficient) 
will  either  be  0 or  1,  or  a constituent,  or  the  sum  of  a part  of  the 
constituents  which  involve  the  symbols  x and  y.  If  A = 0 it  is 
evident  that  E = 0 ; if  A is  a single  constituent,  or  the  sum  of  a 
part  of  the  constituents  involving  x and  y,  E will  be  0.  For  the 
full  development  of  A,  with  respect  to  x and  y,  will  contain  terms 
with  vanishing  coefficients,  and  E is  the  product  of  all  the  co- 
efficients. Hence  when  A = 1,  E\s  equal  to  A,  but  in  other  cases 
E is  equal  to  0.  Similarly,  when  B = 1,  E is  equal  to  B,  but  in 
other  cases  E'  vanishes.  Hence  the  expression  (2)  will  consist  of 
that  part,  if  any  there  be,  of  (1)  in  which  the  coefficients  A,  B 
are  unity.  And  this  reasoning  is  general.  Suppose,  for  instance, 
that  F involved  the  symbols  x,  y,  z,  t,  and  that  it  were  required 
to  eliminate  x and  y.  Then  if  the  development  of  V,  with  re- 
ference to  z and  t,  were 

zt  + xz{\  - t)  + y (1  - z)  t + (1  -£)  (1  - t), 
the  result  sought  would  be 

zt  + (1  - z ) (1  - 1)  = 0, 

this  being  that  portion  of  the  development  of  which  the  co- 
efficients are  unity. 

Hence,  if  from  any  system  of  equations  we  deduce  a single 
equivalent  equation  F=  0,  F satisfying  the  condition 

F(1  - F)  = 0, 

the  ordinary  processes  of  elimination  may  be  entirely  dispensed 
with,  and  the  single  process  of  development  made  to  supply 
their  place. 

8.  It  may  be  that  there  is  no  practical  advantage  in  the  me- 
thod thus  pointed  out,  but  it  possesses  a theoretical  unity  and 
completeness  which  render  it  deserving  of  regard,  and  I shall  ac- 
cordingly devote  a future  chapter  (XIV.)  to  its  illustration.  The 
progress  of  applied  mathematics  has  presented  other  and  signal 
examples  of  the  reduction  of  systems  of  problems  or  equations  to 
the  dominion  of  some  central  but  pervading  law. 


158 


CONDITIONS  OF  A PERFECT  METHOD.  [CHAP.  X. 

9.  It  is  seen  from  what  precedes  that  there  is  one  class  of 
propositions  to  which  all  the  special  appliances  of  the  above  me- 
thods of  preparation  are  unnecessary.  It  is  that  which  is  cha- 
racterized by  the  following  conditions  : 

First,  That  the  propositions  are  of  the  ordinary  kind,  implied 
by  the  use  of  the  copula  is  or  are,  the  predicates  being  particular. 

Secondly,  That  the  terms  of  the  proposition  are  intelligible 
without  the  supposition  of  any  understood  relation  among  the 
elements  which  enter  into  the  expression  of  those  terms. 

Thirdly,  That  the  propositions  are  independent. 

W e may,  if  such  speculation  is  not  altogether  vain,  permit 
ourselves  to  conjecture  that  these  are  the  conditions  which  would 
be  obeyed  in  the  employment  of  language  as  an  instrument  of 
expression  and  of  thought,  by  unerring  beings,  declaring  simply 
what  they  mean,  without  suppression  on  the  one  hand,  and  with- 
out repetition  on  the  other.  Considered  both  in  their  relation 
to  the  idea  of  a perfect  language,  and  in  their  relation  to  the  pro- 
cesses of  an  exact  method,  these  conditions  are  equally  worthy 
of  the  attention  of  the  student. 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


159 


CHAPTER  XI. 

OF  SECONDARY  PROPOSITIONS,  AND  OF  THfe  PRINCIPLES  OF  THEIR 
SYMBOLICAL  EXPRESSION. 

1.  r I ''HE  docti’ine  has  already  been  established  in  Chap,  iv., 
that  every  logical  proposition  may  be  referred  to  one  or 
the  other  of  two  great  classes,  viz.,  Primary  Propositions  and 
Secondary  Propositions.  The  former  of  these  classes  has  been 
discussed  in  the  preceding  chapters  of  this  work,  and  we  are  now 
led  to  the  consideration  of  Secondary  Propositions,  i.  e.  of  Propo- 
sitions concerning,  or  relating  to,  other  propositions  regarded  as 
true  or  false.  The  investigation  upon  which  we  are  entering  will, 
in  its  general  order  and  progress,  resemble  that  which  we  have  al- 
ready conducted.  The  two  inquiries  differ  as  to  the  subjects  of 
thought  which  they  recognise,  not  as  to  the  formal  and  scientific 
laws  which  they  reveal,  or  the  methods  or  processes  which  are 
founded  upon  those  laws.  Probability  would  in  some  measure  fa- 
vour the  expectation  of  such  a result.  It  consists  with  all  that  we 
know  of  the  uniformity  of  Nature,  and  all  that  we  believe  of  the  im- 
mutable const  ancy  of  the  Author  of  Nature,  to  suppose,  that  in  the 
mind,  which  has  been  endowed  with  such  high  capabilities,  not 
only  for  converse  with  surrounding  scenes,  but  for  the  knowledge 
of  itself,  and  for  reflection  upon  the  laws  of  its  own  constitution, 
there  should  exist  a harmony  and  uniformity  not  less  real  than 
that  which  the  study  of  the  physical  sciences  makes  known  to  us. 
Anticipations  such  as  this  are  never  to  be  made  the  primary  rule 
of  our  inquiries,  nor  are  they  in  any  degree  to  divert  us  from 
those  labours  of  patient  research  by  which  we  ascertain  what  is 
the  actual  constitution  of  things  within  the  particular  province 
submitted  to  investigation.  But  when  the  grounds  of  resem- 
blance have  been  properly  and  independently  determined,  it  is 
not  inconsistent,  even  with  purely  scientific  ends,  to  make  that 
resemblance  a subject  of  meditation,  to  trace  its  extent,  and  to 
receive  the  intimations  of  truth,  yet  undiscovered,  which  it  may 


160  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

seem  to  us  to  convey.  The  necessity  of  a final  appeal  to  fact  is 
not  thus  set  aside,  nor  is  the  use  of  analogy  extended  beyond  its 
proper  sphere, — the  suggestion  of  relations  which  independent 
inquiry  must  either  verify  or  cause  to  be  rejected. 

2.  Secondary  Propositions  are  those  which  concern  or  relate  to 
Propositions  considered  as  true  or  false.  The  relations  of  things 
we  express  by  primary  propositions.  But  we  are  able  to  make 
Propositions  themselves  also  the  subject  of  thought,  and  to  ex- 
press our  judgments  concerning  them.  The  expression  of  any 
such  judgment  constitutes  a secondary  proposition.  There  exists 
no  proposition  whatever  of  which  a competent  degree  of  know- 
ledge would  not  enable  us  to  make  one  or  the  other  of  these  two 
assertions,  viz.,  either  that  the  proposition  is  true,  or  that  it  is 
false ; and  each  of  these  assertions  is  a secondary  proposition.  “ It 
is  true  that  the  sun  shines  “It  is  not  true  that  the  planets 
shine  by  their  own  light are  examples  of  this  kind.  In  the 
former  example  the  Proposition  “ The  sun  shines,”  is  asserted  to 
be  true.  In  the  latter,  the  Proposition,  “ The  planets  shine  by 
their  own  light,”  is  asserted  to  be  false.  Secondary  propositions 
also  include  all  judgments  by  which  we  express  a relation  or  de- 
pendence among  propositions.  To  this  class  or  division  we  may 
refer  conditional  propositions,  as,  “If  the  sun  shine  the  day  will 
be  fair.”  Also  most  disjunctive  propositions,  as,  “ Either  the  sun 
will  shine,  or  the  enterprise  will  be  postponed.”  In  the  former 
example  we  express  the  dependence  of  the  truth  of  the  Propo- 
sition, “ The  day  will  be  fair,”  upon  the  truth  of  the  Proposition, 
“ The  sun  will  shine.”  In  the  latter  we  express  a relation  between 
the  two  Propositions,  “ The  sun  will  shine,”  “ The  enterprise  will 
be  postponed,”  implying  that  the  truth  of  the  one  excludes  the 
truth  of  the  other.  To  the  same  class  of  secondary  propositions  we 
must  also  refer  all  those  propositions  which  assert  the  simultaneous 
truth  or  falsehood  of  propositions,  as,  “ It  is  not  true  both  that 
‘ the  sun  will  shine’  and  that  4 the  journey  will  be  postponed.’  ” 
The  elements  of  distinction  which  we  have  noticed  may  even  be 
blended  together  in  the  same  secondary  proposition.  It  may  in- 
volve both  the  disjunctive  element  expressed  by  either , or,  and 
the  conditional  element  expressed  by  if;  in  addition  to  which, 
the  connected  propositions  may  themselves  be  of  a compound 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


161 


character.  If 11  the  sun  shine,”  and  “ leisure  permit,”  then  either 
“ the  enterprise  shall  be  commenced,”  or  “ some  preliminary 
step  shall  be  taken.”  In  this  example  a number  of  propositions 
are  connected  together,  not  arbitrarily  and  unmeaningly,  but  in 
such  a manner  as  to  express  a definite  connexion  between  them,  — a 
connexion  having  reference  to  their  respective  truth  or  falsehood. 
This  combination,  therefore,  according  to  our  definition,  forms 
a Secondary  Proposition. 

The  theory  of  Secondary  Propositions  is  deserving  of  at- 
tentive study,  as  well  on  account  of  its  varied  applications,  as 
for  that  close  and  harmonious  analogy,  already  referred  to,  which 
it  sustains  with  the  theory  of  Primai’y  Propositions.  Upon  each 
of  these  points  I desire  to  offer  a few  further  observations. 

3.  I would  in  the  first  place  remark,  that  it  is  in  the  form  of 
secondary  propositions,  at  least  as  often  as  in  that  of  primary  pro- 
positions, that  the  reasonings  of  ordinary  life  are  exhibited.  The 
discourses,  too,  of  the  moralist  and  the  metaphysician  are  perhaps 
less  often  concerning  things  and  their  qualities,  than  concerning 
principles  and  hypotheses,  concerning  truths  and  the  mutual  con- 
nexion and  relation  of  truths.  The  conclusions  which  our  narrow 
experience  suggests  in  relation  to  the  great  questions  of  morals  and 
society  yet  unsolved,  manifest,  in  more  ways  than  one,  the  limi- 
tations of  their  human  origin ; and  though  the  existence  of  uni- 
versal principles  is  not  to  be  questioned,  the  partial  formulae 
which  comprise  our  knowledge  of  their  application  are  subject 
to  conditions,  and  exceptions,  and  failure.  Thus,  in  those  de- 
partments of  inquiry  which,  from  the  nature  of  their  subject- 
matter,  should  be  the  most  interesting  of  all,  much  of  our  actual 
knowledge  is  hypothetical.  That  there  has  been  a strong  ten- 
dency to  the  adoption  of  the  same  forms  of  thought  in  writers 
on  speculative  philosophy,  will  hereafter  appear.  Hence  the  in- 
troduction of  a general  method  for  the  discussion  of  hypothetical 
and  the  other  varieties  of  secondary  propositions,  will  open  to  us 
a more  interesting  field  of  applications  than  we  have  before  met 
with. 

4.  The  discussion  of  the  theory  of  Secondary  Propositions  is 
in  the  next  place  interesting,  from  the  close  and  remarkable  ana- 
logy which  it  bears  with  the  theory  of  Primary  Propositions.  It 


162  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

will  appear,  that  the  formal  laws  to  which  the  operations  of  the  mind 
are  subject,  are  identical  in  expression  in  both  cases.  The  mathe- 
matical processes  which  are  founded  on  those  laws  are,  therefore, 
identical  also.  Thus  the  methods  which  have  been  investigated 
in  the  former  portion  of  this  work  will  continue  to  be  available 
in  the  new  applications  to  which  we  are  about  to  proceed.  But 
while  the  laws  and  processes  of  the  method  remain  unchanged, 
the  rule  of  interpretation  must  be  adapted  to  new  conditions. 
Instead  of  classes  of  things,  we  shall  have  to  substitute  propo- 
sitions, and  for  the  relations  of  classes  and  individuals,  we  shall 
have  to  consider  the  connexions  of  propositions  or  of  events. 
Still,  between  the  two  systems,  however  differing  in  purport  and 
interpretation,  there  will  be  seen  to  exist  a pervading  harmonious 
relation,  an  analogy  which,  while  it  serves  to  facilitate  the  con- 
quest of  every  yet  remaining  difficulty,  is  of  itself  an  interesting 
subject  of  study,  and  a conclusive  proof  of  that  unity  of  cha- 
racter which  marks  the  constitution  of  the  human  faculties. 

Proposition  I. 

5.  To  investigate  the  nature  of  the  connexion  of  Secondary  Pro- 
positions with  the  idea  of  Time. 

It  is  necessary,  in  entering  upon  this  inquiry,  to  state  clearly 
the  nature  of  the  analogy  which  connects  Secondary  with  Primary 
Propositions. 

Primary  Propositions  express  relations  among  things,  viewed 
as  component  parts  of  a universe  within  the  limits  of  which, 
whether  coextensive  with  the  limits  of  the  actual  universe  or 
not,  the  matter  of  our  discourse  is  confined.  The  relations  ex- 
pressed are  essentially  substantive.  Some,  or  all,  or  none,  of  the 
members  of  a given  class,  are  also  members  of  another  class. 
The  subjects  to  which  primary  propositions  refer — the  relations 
among  those  subjects  which  they  express — are  all  of  the  above 
character. 

But  in  treating  of  secondary  propositions,  we  find  ourselves  con- 
cerned with  another  class  both  of  subjects  and  relations.  For  the 
subjects  with  which  we  have  to  do  are  themselves  propositions,  so 
that  the  question  may  be  asked, — Can  we  regard  these  subjects 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


163 


also  as  things,  and  refer  them,  by  analogy  with  the  previous 
case,  to  a universe  of  their  own  ? Again,  the  relations  among 
these  subject  propositions  are  relations  of  coexistent  truth  or 
falsehood,  not  of  substantive  equivalence.  We  do  not  say,  when 
expressing  the  connexion  of  two  distinct  propositions,  that  the 
one  is  the  other,  but  use  some  such  forms  of  speech  as  the  fol- 
lowing, according  to  the  meaning  which  we  desire  to  convey ; 
“ Either  the  proposition  X is  true,  or  the  proposition  Y is  true 
“ If  the  proposition  X is  true,  the  proposition  Y is  true “ The 
propositions  X and  Y are  jointly  true and  so  on. 

Now,  in  considering  any  such  relations  as  the  above,  we  are 
not  called  upon  to  inquire  into  the  whole  extent  of  their  possible 
meaning  (for  this  might  involve  us  in  metaphysical  questions  of 
causation,  which  are  beyond  the  proper  limits  of  science)  ; but  it 
suffices  to  ascertain  some  meaning  which  they  undoubtedly  pos- 
sess, and  which  is  adequate  for  the  purposes  of  logical  deduction. 
Let  us  take,  as  an  instance  for  examination,  the  conditional  pro- 
position, “If  the  proposition  X is  true,  the  proposition  Y is 
true.”  An  undoubted  meaning  of  this  proposition  is,  that  the 
time  in  which  the  proposition  X is  true,  is  time  in  which  the  pro- 
position Y is  true.  This  indeed  is  only  a relation  of  coexistence, 
and  may  or  may  not  exhaust  the  meaning  of  the  proposition,  but 
it  is  a relation  really  involved  in  the  statement  of  the  proposition, 
and  further,  it  suffices  for  all  the  purposes  of  logical  inference. 

The  language  of  common  life  sanctions  this  view  of  the  es- 
sential connexion  of  secondary  propositions  with  the  notion  of 
time.  Thus  we  limit  the  application  of  a primary  proposition  by 
the  word  “ some,”  but  that  of  a secondary  proposition  by  the 
word  “sometimes.”  To  say,  “ Sometimes  injustice  triumphs,” 
is  equivalent  to  asserting  that  there  are  times  in  which  the  pro- 
position “ Injustice  now  triumphs,”  is  a true  proposition.  There 
are  indeed  propositions,  the  truth  of  which  is  not  thus  limited  to 
particular  periods  or  conjunctures ; propositions  which  are  true 
throughout  all  time,  and  have  received  the  appellation  of  “ eter- 
nal truths.”  The  distinction  must  be  familiar  to  every  reader  of 
Plato  and  Aristotle,  by  the  latter  of  whom,  especially,  it  is  em- 
ployed to  denote  the  contrast  between  the  abstract  verities  of 
science,  such  as  the  propositions  of  geometry  which  are  always 


164  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

true,  and  those  contingent  or  phenomenal  relations  of  things 
which  are  sometimes  true  and  sometimes  false.  But  the  forms  of 
language  in  which  both  kinds  of  propositions  are  expressed  ma- 
nifest a common  dependence  upon  the  idea  of  time ; in  the  one 
case  as  limited  to  some  finite  duration,  in  the  other  as  stretched 
out  to  eternity. 

6.  It  may  indeed  be  said,  that  in  ordinary  reasoning  we  are 
often  quite  unconscious  of  this  notion  of  time  involved  in  the  very 
language  we  are  using.  But  the  remark,  however  just,  only 
serves  to  show  that  we  commonly  reason  by  the  aid  of  words 
and  the  forms  of  a well-constructed  language,  without  attending 
to  the  ulterior  grounds  upon  which  those  very  forms  have  been 
established.  The  course  of  the  present  investigation  will  afford  an 
illustration  of  the  very  same  principle.  I shall  avail  myself  of 
the  notion  of  time  in  order  to  determine  the  laws  of  the  expression 
of  secondary  propositions,  as  well  as  the  laws  of  combination  of 
the  symbols  by  which  they  are  expressed.  But  when  those 
laws  and  those  forms  are  once  determined,  this  notion  of  time 
(essential,  as  I believe  it  to  be,  to  the  above  end)  may  practically 
be  dispensed  with.  We  may  then  pass  from  the  forms  of  com- 
mon language  to  the  closely  analogous  forms  of  the  symbolical 
instrument  of  thought  here  developed,  and  use  its  processes,  and 
interpret  its  results,  without  any  conscious  recognition  of  the  idea 
of  time  whatever. 

Proposition  II. 

7.  To  establish  a system  of  notation  for  the  expression  of 
Secondary  Propositions,  and  to  show  that  the  symbols  which  it 
involves  are  subject  to  the  same  laivs  of  combination  as  the  corres- 
ponding symbols  employed  in  the  expression  of  Primary  Propo- 
sitions. 

Let  us  employ  the  capital  letters  X , Y,  Z,  to  denote  the  ele- 
mentary propositions  concerning  which  we  desire  to  make  some 
assertion  touching  their  truth  or  falsehood,  or  among  which  we 
seek  to  express  some  relation  in  the  form  of  a secondary  propo- 
sition. And  let  us  employ  the  corresponding  small  letters  x,  y,  z, 
considered  as  expressive  of  mental  operations,  in  the  following 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


165 


sense,  viz. : Let  x represent  an  act  of  the  mind  by  which  we  fix 
our  regard  upon  that  portion  of  time  for  which  the  proposition  X 
is  true ; and  let  this  meaning  be  understood  when  it  is  asserted 
that  x denotes  the  time  for  which  the  proposition  X is  true.  Let 
us  further  employ  the  connecting  signs  +,  =,  &c.,  in  the  fol- 

lowing sense,  viz. : Let  x + y denote  the  aggregate  of  those  por- 
tions of  time  for  which  the  propositions  X and  F are  respectively 
true,  those  times  being  entirely  separated  from  each  other.  Si- 
milarly let  x - y denote  that  remainder  of  time  which  is  left  when 
we  take  away  from  the  portion  of  time  for  which  X is  true,  that 
(by  supposition)  included  portion  for  which  Fis  true.  Also,  let 
x = y denote  that  the  time  for  which  the  proposition  X is  true, 
is  identical  with  the  time  for  which  the  proposition  Y is  true. 
We  shall  term  x the  representative  symbol  of  the  proposition  A,  &c. 

From  the  above  definitions  it  will  follow,  that  we  shall 
always  have 

x + y = y + x, 

for  either  member  will  denote  the  same  aggregate  of  time. 

Let  us  further  represent  by  xy  the  performance  in  succession 
of  the  two  operations  represented  by  y and  x,  i.  e.  the  whole 
mental  operation  which  consists  of  the  following  elements,  viz., 
1st,  The  mental  selection  of  that  portion  of  time  for  which  the 
proposition  Y is  true.  2ndly,  The  mental  selection,  out  of  that 
portion  of  time,  of  such  portion  as  it  contains  of  the  time  in 
which  the  proposition  X is  true, — the  result  of  these  successive 
processes  being  the  fixing  of  the  mental  regard  upon  the  whole 
of  that  portion  of  time  for  which  the  propositions  X and  F are 
both  true. 

From  this  definition  it  will  follow,  that  we  shall  always  have 

xy  = yx.  (1) 

For  whether  we  select  mentally,  first  that  portion  of  time  for 
which  the  proposition  F is  true,  then  out  of  the  result  that  con- 
tained portion  for  which  X is  true ; or  first,  that  portion  of  time 
for  which  the  proposition  X is  true,  then  out  of  the  result  that 
contained  portion  of  it  for  which  the  proposition  F is  true  ; we 
shall  arrive  at  the  same  final  result,  viz.,  that  portion  of  time  for 
which  the  propositions  X and  F are  both  true. 


166  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

By  continuing  this  method  of  reasoning  it  may  be  established, 
that  the  laws  of  combination  of  the  symbols  x,  y,  z,  &c.,  in  the 
species  of  interpretation  here  assigned  to  them,  are  identical  in 
expression  with  the  laws  of  combination  of  the  same  symbols,  in 
the  interpretation  assigned  to  them  in  the  first  part  of  this 
treatise.  The  reason  of  this  final  identity  is  apparent.  For  in 
both  cases  it  is  the  same  faculty,  or  the  same  combination  of  fa- 
culties, of  which  we  study  the  operations  ; operations,  the  essen- 
tial character  of  which  is  unaffected,  whether  we  suppose  them  to 
be  engaged  upon  that  universe  of  things  in  which  all  existence 
is  contained,  or  upon  that  whole  of  time  in  which  all  events  are 
realized,  and  to  some  part,  at  least,  of  which  all  assertions, 
truths,  and  propositions,  refer. 

Thus,  in  addition  to  the  laws  above  stated,  we  shall  have  by 
(4),  Chap,  n.,  the  law  whose  expression  is 

x (y  + z)  = xy  + xz  ; (2) 

and  more  particularly  the  fundamental  law  of  duality  (2)  Chap,  n., 
whose  expression  is 

x?  = x,  or,  x (1  - x)  = 0 ; (3) 

a law,  which  while  it  serves  to  distinguish  the  system  of  thought 
in  Logic  from  the  system  of  thought  in  the  science  of  quantity, 
gives  to  the  processes  of  the  former  a completeness  and  a gene- 
rality which  they  could  not  otherwise  possess. 

8.  Again,  as  this  law  (3)  (as  well  as  the  other  laws)  is  satis- 
fied by  the  symbols  0 and  1,  we  are  led,  as  before,  to  inquire 
whether  those  symbols  do  not  admit  of  interpretation  in  the  pre- 
sent system  of  thought.  The  same  course  of  reasoning  which  we 
before  pursued  shows  that  they  do,  and  warrants  us  in  the  two 
following  positions,  viz. : 

1st,  That  in  the  expression  of  secondary  propositions,  0 re- 
presents nothing  in  reference  to  the  element  of  time. 

2nd,  That  in  the  same  system  1 represents  the  universe,  or 
whole  of  time,  to  which  the  discourse  is  supposed  in  any  manner 
to  relate. 

As  in  primary  propositions  the  universe  of  discourse  is  some- 
times limited  to  a small  portion  of  the  actual  universe  of  things, 
and  is  sometimes  co-extensive  with  that  universe ; so  in  secon- 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


167 


dary  propositions,  the  universe  of  discourse  may  be  limited  to  a 
single  day  or  to  the  passing  moment,  or  it  may  comprise  the 
whole  duration  of  time.  It  may,  in  the  most  literal  sense,  be 
“ eternal.”  Indeed,  unless  there  is  some  limitation  expressed  or 
implied  in  the  nature  of  the  discourse,  the  proper  interpretation 
of  the  symbol  1 in  secondary  propositions  is  “ eternity even  as 
its  proper  interpretation  in  the  primary  system  is  the  actually 
existent  universe. 

9.  Instead  of  appropriating  the  symbols  x,  y,  z,  to  the  repre- 
sentation of  the  truths  of  propositions,  we  might  with  equal  pro- 
priety apply  them  to  represent  the  occurrence  of  events.  In  fact, 
the  occurrence  of  an  event  both  implies,  and  is  implied  by,  the 
truth  of  a proposition,  viz.,  of  the  proposition  which  asserts  the 
occurrence  of  the  event.  The  one  signification  of  the  symbol  x 
necessarily  involves  the  other.  It  will  greatly  conduce  to  con- 
venience to  be  able  to  employ  our  symbols  in  either  of  these 
really  equivalent  interpretations  which  the  circumstances  of  a 
problem  may  suggest  to  us  as  most  desirable ; and  of  this  liberty 
I shall  avail  myself  whenever  occasion  requires.  In  problems  of 
pure  Logic  I shall  consider  the  symbols  x,  y,  &c.  as  representing 
elementary  propositions,  among  which  relation  is  expressed  in 
the  premises.  In  the  mathematical  theory  of  probabilities,  which, 
as  before  intimated  (I.  12),  rests  upon  a basis  of  Logic,  and 
which  it  is  designed  to  treat  in  a subsequent  portion  of  this  work, 
I shall  employ  the  same  symbols  to  denote  the  simple  events, 
whose  implied  or  required  frequency  of  occurrence  it  counts 
among  its  elements. 

Proposition  III. 

10.  To  deduce  general  Rules  for  the  expression  of  Secondary 
Propositions. 

In  the  various  inquiries  arising  out  of  this  Proposition,  fulness 
of  demonstration  will  be  the  less  necessary,  because  of  the  exact 
analogy  which  they  bear  with  similar  inquiries  already  completed 
with  reference  to  primary  propositions.  We  shall  first  consider 
the  expression  of  terms ; secondly,  that  of  the  propositions  by 
which  they  are  connected. 


168  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI 

As  1 denotes  the  whole  duration  of  time,  and  x that  portion 
of  it  for  which  the  proposition  X is  true,  1 - x will  denote  that 
portion  of  time  for  which  the  proposition  X is  false. 

Again,  as  xy  denotes  that  portion  of  time  for  which  the  pro- 
positions X and  Y are  both  true,  we  shall,  by  combining  this  and 
the  previous  observation,  be  led  to  the  following  interpretations, 
viz. : 

The  expression  x (1  - y)  will  represent  the  time  during  which 
the  proposition  X is  true,  and  the  proposition  Y false.  The  ex- 
pression (1  - x)  (1  - y)  will  represent  the  time  during  which  the 
propositions  X and  Y are  simultaneously  false. 

The  expression  *(1  - y)  + y ( 1 - x)  will  texpress  the  time 
during  which  either  X is  true  or  Y true,  but  not  both ; for  that 
time  is  the  sum  of  the  times  in  which  they  are  singly  and  exclu- 
sively true.  The  expression  xy  + (1  - «)  (1  - y)  will  express  the 
time  during  which  X and  Y are  either  both  true  or  both  false. 

If  another  symbol  2 presents  itself,  the  same  principles  remain 
applicable.  Thus  xyz  denotes  the  time  in  which  the  propositions 
X,  Y,  and  Z are  simultaneously  true ; ( 1 - .z)  (1  - y)  (1  - 2)  the 
time  in  which  they  are  simultaneously  false;  and  the  sum  of 
these  expressions  would  denote  the  time  in  which  they  are  either 
true  or  false  together. 

The  general  principles  of  interpretation  involved  in  the  above 
examples  do  not  need  any  further  illustrations  or  more  explicit 
statement. 

11.  The  laws  of  the  expression  of  propositions  may  now  be 
exhibited  and  studied  in  the  distinct  cases  in  which  they  present 
themselves.  There  is,  however,  one  principle  of  fundamental 
importance  to  which  I wish  in  the  first  place  to  direct  attention. 
Although  the  principles  of  expression  which  have  been  laid  down 
are  perfectly  general,  and  enable  us  to  limit  our  assertions  of  the 
truth  or  falsehood  of  propositions  to  any  particular  portions  of 
that  whole  of  time  (whether  it  be  an  unlimited  eternity,  or  a pe- 
riod whose  beginning  and  whose  end  are  definitely  fixed,  or  the 
passing  moment)  which  constitutes  the  universe  of  our  discourse, 
yet,  in  the  actual  procedure  of  human  reasoning,  such  limitation 
is  not  commonly  employed.  When  we  assert  that  a proposition 
is  true,  we  generally  mean  that  it  is  true  throughout  the  whole 


CHAP.  XI.]  OF  SECONDARY  PROPOSITIONS.  169 

duration  of  the  time  to  which  our  discourse  refers ; and  when  dif- 
ferent assertions  of  the  unconditional  truth  or  falsehood  of  propo- 
sitions are  jointly  made  as  the  premises  of  a logical  demonstration, 
it  is  to  the  same  universe  of  time  that  those  assertions  are  re- 
ferred, and  not  to  particular  and  limited  parts  of  it.  In  that 
necessary  matter  which  is  the  object  or  field  of  the  exact  sciences 
every  assertion  of  a truth  may  be  the  assertion  of  an  “ eternal 
truth.”  In  reasoning  upon  transient  phienomena,  (as  of  some 
social  conjuncture)  each  assertion  may  be  qualified  by  an  imme- 
diate reference  to  the  present  time,  “ Now.”  But  in  both  cases, 
unless  there  is  a distinct  expression  to  the  contrary,  it  is  to  the 
same  period  of  duration  that  each  separate  proposition  relates. 
The  cases  which  then  arise  for  our  consideration  are  the  fol- 
lowing : 

1st.  To  express  the  Proposition , “ The  proposition  X is  true'' 

We  are  here  required  to  express  that  within  those  limits  of 
time  to  which  the  matter  of  our  discourse  is  confined  the  propo- 
sition X is  true.  Now  the  time  for  which  the  proposition  X is 
true  is  denoted  by  x,  and  the  extent  of  time  to  which  our  dis- 
course refers  is  represented  by  1.  Hence  we  have 

x = 1 (4) 

as  the  expression  required. 

2nd.  To  express  the  Proposition,  11  The  proposition  X is 
false." 

We  are  here  to  express  that  within  the  limits  of  time  to  which 
our  discourse  relates,  the  proposition  X is  false ; or  that  within 
those  limits  there  is  no  portion  of  time  for  which  it  is  true.  Now 
the  portion  of  time  for  which  it  is  true  is  x.  Hence  the  required 
equation  will  be 

* = 0.  (5) 

This  result  might  also  be  obtained  by  equating  to  the  whole  du- 
ration of  time  1 , the  expression  for  the  time  during  which  the 
proposition  X is  false,  viz.,  1 - x.  This  gives 

1 - a?  = 1, 

whence  x = 0. 

3rd.  To  express  the  disjunctive  Proposition,  “ Either  the  pro- 


170  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

position  X is  true  or  the  proposition  Y is  true it  being  thereby 
implied  that  the  said  propositions  are  mutually  exclusive,  that  is  to 
say,  that  one  only  of  them  is  true. 

The  time  for  which  either  the  proposition  X is  true  or  the 
proposition  Y is  true,  but  not  both,  is  represented  by  the  ex- 
pression x - y)  + y - x).  Hence  we  have 

x(\-y)+y(l-x)  = \,  (6) 

for  the  equation  required. 

If  in  the  above  Proposition  the  particles  either,  or,  are  sup- 
posed not  to  possess  an  absolutely  disjunctive  power,  so  that  the 
possibility  of  the  simultaneous  truth  of  the  propositions  X and  Y 
is  not  excluded,  we  must  add  to  the  first  member  of  the  above 
equations  the  term  xy.  We  shall  thus  have 

xy  + x{\-y)  + (\-x)y=\, 

or  x + (1  - x)  y = 1.  ^ ' 

4th.  To  express  the  conditional  Proposition,  “ If  the  propo- 
sition Y is  true,  the  proposition  X is  true.” 

Since  whenever  the  proposition  Y is  true,  the  proposition  X 
is  true,  it  is  necessary  and  sufficient  here  to  express,  that  the  time 
in  which  the  proposition  Y is  true  is  time  in  which  the  propo- 
sition X is  true ; that  is  to  say,  that  it  is  some  indefinite  portion 
of  the  whole  time  in  which  the  proposition  X is  true.  Now  the 
time  in  which  the  proposition  Y is  true  is  y,  and  the  whole  time 
in  which  the  proposition  X is  true  is  x.  Let  v be  a symbol  of 
time  indefinite,  then  will  vx  represent  an  indefinite  portion  of  the 
whole  time  x.  Accordingly,  we  shall  have 

y - vx 

as  the  expression  of  the  proposition  given. 

1 2.  When  v is  thus  regarded  as  a symbol  of  time  indefinite, 
vx  may  be  understood  to  represent  the  whole,  or  an  indefinite 
part,  or  no  part,  of  the  whole  time  x ; for  any  one  of  these  mean- 
ings may  be  realized  by  a particular  determination  of  the  arbitrary 
symbol  v.  Thus,  if  v be  determined  to  represent  a time  in  which 
the  whole  time  x is  included,  vx  will  represent  the  whole  time  x. 
If  v be  determined  to  represent  a time,  some  part  of  which  is  in- 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


171 


eluded  in  the  time  x,  but  which  does  not  fill  up  the  measure  of 
that  time,  vx  will  represent  a part  of  the  time  x.  If,  lastly,  v is 
determined  to  represent  a time,  of  which  no  part  is  common  with 
any  part  of  the  time  x,  vx  will  assume  the  value  0,  and  will  be 
equivalent  to  “no  time,”  or  “never.” 

Now  it  is  to  be  observed  that  the  proposition,  “ If  Y is  true, 
X is  true,”  contains  no  assertion  of  the  truth  of  either  of  the 
propositions  X and  Y.  It  may  equally  consist  with  the  suppo- 
sition that  the  truth  of  the  proposition  Y is  a condition  indis- 
pensable to  the  truth  of  the  proposition  X,  in  which  case  we 
shall  have  v = 1 ; or  with  the  supposition  that  although  Y ex- 
presses a condition  which,  when  realized,  assures  us  of  the  truth 
of  X , yet  X may  be  true  without  implying  the  fulfilment  of  that 
condition,  in  which  case  v denotes  a time,  some  part  of  which  is 
contained  in  the  whole  time  x ; or,  lastly,  with  the  supposition 
that  the  proposition  Y is  not  true  at  all,  in  which  case  v repre- 
sents some  time,  no  part  of  which  is  common  with  any  part  of 
the  time  x.  All  these  cases  are  involved  in  the  general  suppo- 
sition that  v is  a symbol  of  time  indefinite. 

5th.  To  express  a proposition  in  which  the  conditional  and  the 
disjunctive  characters  both  exist. 

The  general  form  of  a conditional  proposition  is,  “ If  Y is 
true,  X is  true,”  and  its  expression  is,  by  the  last  section,  y = vx. 
We  may  properly,  in  analogy  with  the  usage  which  has  been  es- 
tablished in  primary  propositions,  designate  Y and  X as  the 
terms  of  the  conditional  proposition  into  which  they  enter ; and 
we  may  further  adopt  the  language  of  the  ordinary  Logic,  which 
designates  the  term  Y,  to  which  the  particle  //"is  attached,  the 
“antecedent”  of  the  proposition,  and  the  term  X the  “conse- 
quent.” 

Now  instead  of  the  terms,  as  in  the  above  case,  being  simple 
propositions,  let  each  or  either  of  them  be  a disjunctive  propo- 
sition involving  different  terms  connected  by  the  particles  either, 
or,  as  in  the  following  illustrative  examples,  in  which  X,  Y,  Z, 
&c.  denote  simple  propositions. 

1st.  If  either  X is  true  or  Y is  true,  then  Z is  true. 

2nd.  If  X is  true,  then  either  Y is  true  or  Z true. 


172  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

3rd.  If  either  X is  true  or  Y is  true,  then  either  Z and  W 
are  both  true,  or  they  are  both  false. 

It  is  evident  that  in  the  above  cases  the  relation  of  the  ante- 
cedent to  the  consequent  is  not  affected  by  the  circumstance  that 
one  of  those  terms  or  both  are  of  a disjunctive  character.  Ac- 
cordingly it  is  only  necessary  to  obtain,  in  conformity  with  the 
principles  already  established,  the  proper  expressions  for  the  ante- 
cedent and  the  consequent,  to  affect  the  latter  with  the  indefinite 
symbol  v,  and  to  equate  the  results.  Thus  for  the  propositions 
above  stated  we  shall  have  the  respective  equations, 

1st.  «(1  - y)  + (1  -x)y  = vz. 

2nd.  x = v [y  (1  - z)  + z (1  - y) } . 

3rd.  x (1  - y)  + y (1  - x)  = v [zw  + (1  - z)  (1  -w)). 

The  rule  here  exemplified  is  of  general  application. 

Cases  in  which  the  disjunctive  and  the  conditional  elements 
enter  in  a manner  different  from  the  above  into  the  expression  of 
a compound  proposition,  are  conceivable,  but  I am  not  aware  that 
they  are  ever  presented  to  us  by  the  natural  exigencies  of  human 
reason,  and  I shall  therefore  refrain  from  any  discussion  of  them. 
No  serious  difficulty  will  arise  from  this  omission,  as  the  general 
principles  which  have  formed  the  basis  of  the  above  applications 
are  perfectly  general,  and  a slight  effort  of  thought  will  adapt 
them  to  any  imaginable  case. 

13.  In  the  laws  of  expression  above  stated  those  of  interpre- 
tation are  implicitly  involved.  The  equation 

x - 1 

must  be  understood  to  express  that  the  proposition  X is  true ; 
the  equation 

x = 0, 

that  the  proposition  X is  false.  The  equation 

xy  = 1 

will  express  that  the  propositions  X and  Y are  both  true  toge- 
ther ; and  the  equation 

xy  = 0 

that  they  are  not  both  together  true. 


CHAP.  XI.] 


OF  SECONDARY  PROPOSITIONS. 


173 


In  like  manner  the  equations 

x{\-y) +y{\- x)  = \, 

- y)  + y(  i - ®)  = o, 

will  respectively  assert  the  truth  and  the  falsehood  of  the  disjunc- 
tive Proposition,  “Either  X is  true  or  Y is  true.”  The  equa- 
tions 

y = vx 

y = v(l  - x) 

will  respectively  express  the  Propositions,  “If  the  proposition  Y 
is  true,  the  proposition  X is  true.”  “ If  the  proposition  Y is 
true,  the  proposition  X is  false.” 

Examples  will  frequently  present  themselves,  in  the  suc- 
ceeding chapters  of  this  work,  of  a case  in  which  some  terms  of  a 
particular  member  of  an  equation  are  affected  by  the  indefinite 
symbol  v,  and  others  not  so  affected.  The  following  instance 
will  serve  for  illustration.  Suppose  that  we  have 
y = xz  + vx  (1  - z). 

Here  it  is  implied  that  the  time  for  which  the  proposition  Y is 
true  consists  of  all  the  time  for  which  X and  Z are  together  true, 
together  with  an  indefinite  portion  of  the  time  for  which  X is 
true  and  Z false.  From  this  it  may  be  seen,  1st,  That  if  T~is 
true,  either  X and  Z are  together  true,  or  X is  true  and  Z false ; 
2ndly,  If  X and  Z are  together  true,  Y is  true.  The  latter  of 
these  may  be  called  the  reverse  interpretation,  and  it  consists  in 
taking  the  antecedent  out  of  the  second  member,  and  the  conse- 
quent from  the  first  member  of  the  equation.  The  existence  of 
a term  in  the  second  member,  whose  coefficient  is  unity,  renders 
this  latter  mode  of  interpretation  possible.  The  general  principle 
which  it  involves  may  be  thus  stated  : 

14.  Principle. — Any  constituent  term  or  terms  in  a particular 
member  of  an  equation  which  have  for  their  coefficient  unity,  may 
be  taken  as  the  antecedent  of  a proposition,  of  which  all  the  terms 
in  the  other  member  form  the  consequent. 

Thus  the  equation 

y = xz  + vx  (1  - z)  + (1  - x)  (1  - z) 
would  have  the  following  interpretations  : 


174 


OF  SECONDARY  PROPOSITIONS. 


[CHAP.  XI. 

Direct  Interpretation. — If  the  proposition  Y is  true,  then 
either  X and  Z are  true,  or  X is  true  and  Z false,  or  X and  Z 
are  both  false. 

Reverse  Interpretation. — If  either  X and  Z are  true,  or 
X and  Z are  false,  5r  is  true. 

The  aggregate  of  these  partial  interpretations  will  express 
the  whole  significance  of  the  equation  given. 

15.  We  may  here  call  attention  again  to  the  remark,  that 
although  the  idea  of  time  appears  to  be  an  essential  element  in 
the  theory  of  the  interpretation  of  secondary  propositions,  it  may 
practically  be  neglected  as  soon  as  the  laws  of  expression  and  of 
interpretation  are  definitely  established.  The  forms  to  which 
those  laws  give  rise  seem,  indeed,  to  correspond  with  the  forms  of 
a perfect  language.  Let  us  imagine  any  known  or  existing  lan- 
guage freed  from  idioms  and  divested  of  superfluity,  and  let  us 
express  in  that  language  any  given  proposition  in  a manner  the 
most  simple  and  literal, — the  most  in  accordance  with  those 
principles  of  pure  and  universal  thought  upon  which  all  languages 
are  founded,  of  which  all  bear  the  manifestation,  but  from  which 
all  have  more  or  less  departed.  The  transition  from  such  a lan- 
guage to  the  notation  of  analysis  would  consist  of  no  more  than 
the  substitution  of  one  set  of  signs  for  another,  without  essential 
change  either  of  form  or  character.  For  the  elements,  whether 
things  or  propositions,  among  which  relation  is  expressed,  we 
should  substitute  letters ; for  the  disjunctive  conjunction  we 
should  write  + ; for  the  connecting  copula  or  sign  of  relation,  we 
should  write  =.  This  analogy  I need  not  pursue.  Its  reality 
and  completeness  will  be  made  more  apparent  from  the  study  of 
those  forms  of  expression  which  will  present  themselves  in  sub- 
sequent applications  of  the  present  theory,  viewed  in  more  imme- 
diate comparison  with  that  imperfect  yet  noble  instrument  of 
thought — the  English  language. 

16.  Upon  the  general  analogy  between  the  theory  of  Primary 
and  that  of  Secondary  Propositions,  I am  desirous  of  adding  a 
few  remarks  before  dismissing  the  subject  of  the  present  chapter. 

We  might  undoubtedly,  have  established  the  theory  of  Pri- 
mary Propositions  upon  the  simple  notion  of  space,  in  the  same 


CHAP.  XI.]  OF  SECONDARY  PROPOSITIONS.  175 

way  as  that  of  secondary  propositions  has  been  established  upon 
the  notion  of  time.  Perhaps,  had  this  been  done,  the  analogy 
which  we  are  contemplating  would  have  been  in  somewhat  closer 
accordance  with  the  view  of  those  who  regard  space  and  time 
as  merely  41  forms  of  the  human  understanding,”  conditions  of 
knowledge  imposed  by  the  very  constitution  of  the  mind  upon 
all  that  is  submitted  to  its  apprehension.  But  this  view,  while 
on  the  one  hand  it  is  incapable  of  demonstration,  on  the  other 
hand  ties  us  down  to  the  recognition  of  44  place,”  to  ttov,  as  an 
essential  category  of  existence.  The  question,  indeed,  whether 
it  is  so  or  not,  lies,  I apprehend,  beyond  the  reach  of  our  faculties ; 
but  it  may  be,  and  I conceive  has  been,  established,  that  the 
formal  processes  of  reasoning  in  primary  propositions  do  not  re- 
quire, as  an  essential  condition,  the  manifestation  in  space  of  the 
things  about  which  we  reason ; that  they  would  remain  appli- 
cable, with  equal  strictness  of  demonstration,  to  forms  of  exis- 
tence, if  such  there  be,  which  lie  beyond  the  realm  of  sensible 
extension.  It  is  a fact,  perhaps,  in  some  degree  analogous  to  this, 
that  we  are  able  in  many  known  examples  in  geometry  and  dy- 
namics, to  exhibit  the  formal  analysis  of  problems  founded  upon 
some  intellectual  conception  of  space  different  from  that  which  is 
presented  to  us  by  the  senses,  or  which  can  be  realized  by  the 
imagination.*  I conceive,  therefore,  that  the  idea  of  space  is  not 


* Space  Is  presented  to  us  in  perception,  as  possessing  the  three  dimensions 
of  length,  breadth,  and  depth.  But  in  a large  class  of  problems  relating  to  the 
properties  of  curved  surfaces,  the  rotations  of  solid  bodies  around  axes,  the  vi- 
brations of  elastic  media,  &c.,  this  limitation  appears  in  the  analytical  investi- 
gation to  be  of  an  arbitrary  character,  and  if  attention  were  paid  to  the  processes 
of  solution  alone,  no  reason  could  be  discovered  why  space  should  not  exist  in 
four  or  in  any  greater  number  of  dimensions.  The  intellectual  procedure  in 
the  imaginary  world  thus  suggested  can  be  apprehended  by  the  clearest  light  of 
analogy. 

The  existence  of  space  in  three  dimensions,  and  the  views  thereupon  of  the 
religious  and  philosophical  mind  of  antiquity,  are  thus  set  forth  by  Aristotle: — 
Mfyidoc  Si  to  fiiv  zip  iv,  ypappp,  to  5’  ziri  Svo  zttitzSov,  to  S'  ztt'l  Tpia  <ruipa' 
Kai  7rapa  Tavra  ovk  zstiv  aUo  pzyzOug,  Sea  to  rpid  Tcdvra  ei veil  Kai  to  Tpig 
KavTi).  Kddarrzp  yap  <pa<n  icai  oi  Hvdayopziot,  to  Trav  Kai  rd  iravra  t oig  rpiaiv 
wpiarai.  TzXzvtt)  yap  Kai  ptaov  Kai  apxv  T°v  apidpbv  z\zi  t'ov  too  iravTog' 
ravra  Si  t'ov  Trjg  rpidSog.  A 10  irapa  rrjg  ipvrrzwg  ziXptporzg  uxnrzp  vopovg  ZKtivrjg, 
Kai  npograg  ayioTtiag  xpuptda  twv  dziuv  Tip  dpiQpip  ToiiTtp. — De  Calo,  1. 


176  OF  SECONDARY  PROPOSITIONS.  [CHAP.  XI. 

essential  to  the  development  of  a theory  of  primary  propositions, 
but  am  disposed,  though  desiring  to  speak  with  diffidence  upon 
a question  of  such  extreme  difficulty,  to  think  that  the  idea  of 
time  is  essential  to  the  establishment  of  a theory  of  secondary 
propositions.  There  seem  to  be  grounds  for  thinking,  that 
without  any  change  in  those  faculties  which  are  concerned  in 
reasoning , the  manifestation  of  space  to  the  human  mind  might 
have  been  different  from  what  it  is,  but  not  (at  least  the  same) 
grounds  for  supposing  that  the  manifestation  of  time  could  have 
been  otherwise  than  we  perceive  it  to  be.  Dismissing,  however, 
these  speculations  as  possibly  not  altogether  free  from  presump- 
tion, let  it  be  affirmed  that  the  real  ground  upon  which  the 
symbol  1 represents  in  primary  propositions  the  universe  of 
things,  and  not  the  space  they  occupy,  is,  that  the  sign  of 
identity  = connecting  the  members  of  the  corresponding  equa- 
tions, implies  that  the  things  which  they  represent  are  identical, 
not  simply  that  they  are  found  in  the  same  portion  of  space. 
Let  it  in  like  manner  be  affirmed,  that  the  reason  why  the  symbol 
1 in  secondary  propositions  represents,  not  the  universe  of  events, 
but  the  eternity  in  whose  successive  moments  and  periods  they 
are  evolved,  is,  that  the  same  sign  of  identity  connecting  the 
logical  members  of  the  corresponding  equations  implies,  not  that 
the  events  which  those  members  represent  are  identical,  but  that 
the  times  of  their  occurrence  are  the  same.  These  reasons  appear 
to  me  to  be  decisive  of  the  immediate  question  of  interpretation.  In 
a former  treatise  on  this  subject  (Mathematical  Analysis  of  Logic, 
p.  49),  following  the  theory  of  Wallis  respecting  the  Reduction 
of  Hypothetical  Propositions,  I was  led  to  interpret  the  symbol  1 
in  secondary  propositions  as  the  universe  of  “ cases”  or  “ conjunc- 
tures of  circumstances but  this  view  involves  the  necessity  of  a 
definition  of  what  is  meant  by  a “ case,”  or  “ conjuncture  of 
circumstances and  it  is  certain,  that  whatever  is  involved  in 
the  term  beyond  the  notion  of  time  is  alien  to  the  objects,  and 
restrictive  of  the  processes,  of  formal  Logic. 


CHAP.  XI!.]  METHODS  IN  SECONDARY  PROPOSITIONS. 


177 


CHAPTER  XII. 

OF  THE  METHODS  AND  PROCESSES  TO  BE  ADOPTED  IN  THE  TREAT- 
MENT OF  SECONDARY  PROPOSITIONS. 

1.  TT  has  appeared  from  previous  researches  (XI.  7)  that  the 
laws  of  combination  of  the  literal  symbols  of  Logic  are  the 
same,  whether  those  symbols  are  employed  in  the  expression  of 
primary  or  in  that  of  secondary  propositions,  the  sole  existing 
difference  between  the  two  cases  being  a difference  of  interpre- 
tation. It  has  also  been  established  (V.  6),  that  whenever  dis- 
tinct systems  of  thought  and  interpretation  are  connected  with 
the  same  system  of  formal  laws,  i.  e.,  of  laws  relating  to  the  com- 
bination and  use  of  symbols,  the  attendant  processes,  intermediate 
between  the  expression  of  the  primary  conditions  of  a problem 
and  the  interpretation  of  its  symbolical  solution,  are  the  same  in 
both.  Hence,  as  between  the  systems  of  thought  manifested  in 
the  two  forms  of  primary  and  of  secondary  propositions,  this  com- 
munity of  formal  law  exists,  the  processes  which  have  been  es- 
tablished and  illustrated  in  our  discussion  of  the  former  class  of 
propositions  will,  without  any  modification,  be  applicable  to  the 
latter. 

2.  Thus  the  laws  of  the  two  fundamental  processes  of  elimi- 
nation and  development  are  the  same  in  the  system  of  secondary 
as  in  the  system  of  primary  propositions.  Again,  it  has  been 
seen  (Chap.  vi.  Prop.  2)  how,  in  primary  propositions,  the  inter- 
pretation of  any  proposed  equation  devoid  of  fractional  forms 
may  be  effected  by  developing  it  into  a series  of  constituents,  and 
equating  to  0 every  constituent  whose  coefficient  does  not  vanish. 
To  the  equations  of  secondary  propositions  the  same  method  is 
applicable,  and  the  interpreted  result  to  which  it  finally  conducts 
us  is,  as  in  the  former  case  (VI.  6),  a system  of  co-existent  denials. 
But  while  in  the  former  case  the  force  of  those  denials  is  ex- 
pended upon  the  existence  of  certain  classes  of  things,  in  the 
latter  it  relates  to  the  truth  of  certain  combinations  of  the  elc- 


178  METHODS  IN  SECONDARY  PROPOSITIONS.  [CHAP.  XII. 

mentary  propositions  involved  in  the  terms  of  the  given  premises. 
And  as  in  primary  propositions  it  was  seen  that  the  system  of 
denials  admitted  of  conversion  into  various  other  forms  of  propo- 
sitions (VI.  7),  &c.,  such  conversion  will  be  found  to  be  possible 
here  also,  the  sole  difference  consisting  not  in  the  forms  of  the 
equations,  but  in  the  nature  of  their  interpretation. 

3.  Moreover,  as  in  primary  propositions,  we  can  find  the  ex- 
pression of  any  element  entering  into  a system  of  equations,  in 
terms  of  the  remaining  elements  (VI.  10),  or  of  any  selected 
number  of  the  remaining  elements,  and  interpret  that  expression 
into  a logical  inference,  the  same  object  can  be  accomplished  by 
the  same  means,  difference  of  interpretation  alone  excepted,  in 
the  system  of  secondary  propositions.  The  elimination  of  those 
elements  which  we  desire  to  banish  from  the  final  solution,  the 
reduction  of  the  system  to  a single  equation,  the  algebraic  solu- 
tion and  the  mode  of  its  development  into  an  interpretable  form, 
differ  in  no  respect  from  the  corresponding  steps  in  the  discussion 
of  primary  propositions. 

To  remove,  however,  any  possible  difficulty,  it  may  be  de- 
sirable to  collect  under  a general  Rule  the  different  cases  which 
present  themselves  in  the  treatment  of  secondary  propositions. 

Rule. — Express  symbolically  the  given  propositions  (XI.  11). 

Eliminate  separately  from  each  equation  in  which  it  is  found  the 
indefinite  symbol  v (VII.  5). 

Eliminate  the  remaining  symbols  ivliich  it  is  desired  to  banish 
from  the  final  solution : alivays  before  elimination  reducing  to  a 
single  equation  those  equations  in  ivliich  the  symbol  or  symbols  to 
be  eliminated  are  found  (VIII.  7).  Collect  the  resulting  equa- 
tions into  a single  equation  V = 0. 

Then  proceed  according  to  the  particular  form  in  ivhich  it  is 
desired  to  express  the  final  relation,  as — 

1st.  If  in  the  form  of  a.  denial , or  system  of  denials,  develop  the 
function  V,  and  equate  to  0 all  those  constituents  ichose  coefficients 
do  not  vanish. 

2ndly.  If  in  the  form  of  a disjunctive  proposition,  equate  to  1 
the  sum  of  those  constituents  ivhose  coffiecients  vanish. 

3rdly.  If  in  the  form  of  a conditional  proposition  having  a,  sim- 


CHAP.  XII.]  METHODS  IN  SECONDARY  PROPOSITIONS. 


179 


pie  element , as  x or  1 - x,  for  its  antecedent , determine  the  alge- 
braic expression  of  that  element,  and  develop  that  expression. 

4thly.  If  in  the  form  of  a conditional  proposition  having  a 
compound  expression,  as  xy,  xy  + (1  - x)  (1  - y),  8fc.,for  its  ante- 
cedent, equate  that  expression  to  a new  symbol  t,  and  determine  t 
as  a developed  function  of  the  symbols  which  are  to  appear  in  the 
consequent,  either  by  ordinary  methods  or  by  the  special  method 

(IX.  9). 

5thly.  Interpret  the  results  by  (XI.  13,  14). 

If  it  only  be  desired  to  ascertain  whether  a particular  elemen- 
tary proposition  x is  true  or  false , we  must  eliminate  all  the  sym- 
bols but  x ; then  the  equation  x-  1 will  indicate  that  the  proposition 
is  true,  x = 0 that  it  is  false,  0 = 0 that  the  premises  are  insufficient 
to  determine  whether  it  is  true  or  false. 

4.  Ex.  1. — The  following  prediction  is  made  the  subject  of  a 
curious  discussion  in  Cicero’s  fragmentary  treatise,  De  Fato  : — 
“ Si  quis  (Fabius)  natus  est  oriente  Canicula,  is  in  mari  non  mo- 
rietur.”  I shall  apply  to  it  the  method  of  this  chapter.  Let  y 
represent  the  proposition,  “ Fabius  was  born  at  the  rising  of  the 
dogstar;”  x the  proposition,  “Fabius  will  die  in  the  sea.” 
In  saying  that  x represents  the  proposition,  “Fabius,  &c.,”  it  is 
only  meant  that  x is  a symbol  so  appropriated  (XI.  7)  to  the 
above  proposition,  that  the  equation  x = 1 declares,  and  the  equa- 
tion x = 0 denies,  the  truth  of  that  proposition.  The  equation 
we  have  to  discuss  will  be 


y = »(!-*). 


0) 


And,  first,  let  it  be  required  to  reduce  the  given  proposition  to  a 
negation  or  system  of  negations  (XII.  3).  We  have,  on  trans- 
position, 

y — v ( 1 - x')  = 0. 


Eliminating  v, 


y [y  -0  - a?)}  = 0, 


or, 


y-y(  1 -x)  = 0. 


or, 


yx  = 0. 


(2) 


The  interpretation  of  this  result  is  : — “ It  is  not  true  that  Fabius 
was  born  at  the  rising  of  the  dogstar,  and  will  die  in  the  sea.” 


180  METHODS  IN  SECONDARY  PROPOSITIONS.  [CHAP.  XII. 

Cicero  terms  this  form  of  proposition,  “ Conjunctio  ex  repug- 
nantibus and  he  remarks  that  Chrysippus  thought  in  this  way 
to  evade  the  difficulty  which  he  imagined  to  exist  in  contingent 
assertions  respecting  the  future  : “ Hoc  loco  Chrysippus  aestuans 
falli  sperat  Chaldajos  casterosque  divinos,  neque  eos  usuros  esse 
conjunctionibus  ut  ita  sua  percepta  pronuntient : Si  quis  natus 
est  oriente  Canicula  is  in  mari  non  morietur ; sed  potius  ita  dicant: 
Hon  et  natus  est  quis  oriente  Canicula,  et  in  mari  morietur. 
O licentiam  jocularem ! . . . . Multa  genera  sunt  enuntiandi,  nec 
ullum  distortius  quam  hoc  quo  Chrysippus  sperat  Chaldteos  con- 
tentos  Stoicorum  causa  fore.” — Cic.  De  Fato,  7,  8. 

5.  To  reduce  the  given  proposition  to  a disjunctive  form. 
The  constituents  not  entering  into  the  first  member  of  (2)  are 

x(\  - y ),  (1  -x)y,  (1  - *)  (1  -y). 

Whence  we  have 

y{\  -tc)  + x(l  -y)  + (l-«)(l  - y)  = 1.  (3) 

The  interpretation  of  which  is  : — Either  Fabius  was  born  at  the 
rising  of  the  dog  star,  and  icill  not  perish  in  the  sea  ; or  he  teas  not 
born  at  the  rising  of  the  dog  star,  and  will  perish  in  the  sea;  or  he 
was  not  born  at  the  rising  of  the  dogstar,  and  will  not  perish  in 
the  sea. 

In  cases  like  the  above,  however,  in  which  there  exist  consti- 
tuents differing  from  each  other  only  by  a single  factor,  it  is,  as 
we  have  seen  (VII.  15),  most  convenient  to  collect  such  consti- 
tuents into  a single  term.  If  we  thus  connect  the  first  and  third 
terms  of  (3),  we  have 

(\  - y') x + \ ~ x = 1 ; 

and  if  we  similarly  connect  the  second  and  third,  we  have 

y{  i -*)  + i -y  = i- 

These  forms  of  the  equation  severally  give  the  interpretations — 
Either  Fabius  ivas  not  born  under  the  dogstar , and  will  die  in 
the  sea,  or  he  ivill  not  die  in  the  sea. 

Either  Fabius  was  born  under  the  dogstar , and  will  not  die  in 
the  sea,  or  he  was  not  born  under  the  dogstar. 


CHAP.  XII.]  METHODS  IN  SECONDARY  PROPOSITIONS. 


181 


It  is  evident  that  these  interpretations  are  strictly  equivalent 
to  the  former  one. 

Let  us  ascertain,  in  the  form  of  a conditional  proposition,  the 
consequences  which  flow  from  the  hypothesis,  that  “ Fabius  will 
perish  in  the  sea.” 

In  the  equation  (2),  which  expresses  the  result  of  the  elimi- 
nation of  v from  the  original  equation,  we  must  seek  to  determine 
x as  a function  of  y. 

We  have 

0 0 x 

x = - = 0 y + - (1-i/)  on  expansion, 


or, 


* = o c1  ~y)> 


the  interpretation  of  which  is, — If  Fabius  shall  die  in  the  sea , he 
was  not  born  at  the  rising  of  the  dog  star. 

These  examples  serve  in  some  measure  to  illustrate  the  con- 
nexion which  has  been  established  in  the  previous  sections  be- 
tween primary  and  secondary  propositions,  a connexion  of  which 
the  two  distinguishing  features  are  identity  of  process  and  analogy 
of  interpretation. 

6.  Ex.  2. — There  is  a remarkable  argument  in  the  second 
book  of  the  Republic  of  Plato,  the  design  of  which  is  to  prove 
the  immutability  of  the  Divine  Nature.  It  is  a very  fine  example 
both  of  the  careful  induction  from  fatpiliar  instances  by  which 
Plato  arrives  at  general  principles,  and  of  the  clear  and  connected 
logic  by  which  he  deduces  from  them  the  particular  inferences 
which  it  is  his  object  to  establish.  The  argument  is  contained 
in  the  following  dialogue  : 

“ Must  not  that  which  departs  from  its  proper  form  be 
changed  either  by  itself  or  by  another  thing  ? Necessarily  so. 
Are  not  things  which  are  in  the  best  state  least  changed  and  dis- 
turbed, as  the  body  by  meats  and  drinks,  and  labours,  and  every 
species  of  plant  by  heats  and  winds,  and  such  like  affections  ? Is 
not  the  healthiest  and  strongest  the  least  changed  ? Assuredly. 
And  does  not  any  trouble  from  without  least  disturb  and  change 
that  soul  which  is  strongest  and  wisest?  And  as  to  all  made 
vessels,  and  furnitures,  and  garments,  according  to  the  same 


182  METHODS  IN  SECONDARY  PROPOSITIONS.  [CHAP.  XII. 

principle,  are  not  those  which  are  well  wrought,  and  in  a good 
condition,  least  changed  by  time  and  other  accidents  ? Even  so. 
And  whatever  is  in  a right  state,  either  by  nature  or  by  art,  or 
by  both  these,  admits  of  the  smallest  change  from  any  other 
thing.  So  it  seems.  But  God  and  things  divine  are  in  every 
sense  in  the  best  state.  Assuredly.  In  this  xvay,  then,  God 
should  least  of  all  bear  many  forms  ? Least,  indeed,  of  all. 
Again,  should  He  transform  and  change  Himself?  Manifestly  He 
must  do  so,  if  He  is  changed  at  all.  Changes  He  then  Himself  to 
that  which  is  more  good  and  fair,  or  to  that  which  is  worse  and 
baser?  Necessarily  to  the  worse,  if  he  be  changed.  For  never 
shall  we  say  that  God  is  indigent  of  beauty  or  of  virtue.  You 
speak  most  rightly,  said  I,  and  the  matter  being  so,  seems  it  to 
you,  O Adimantus,  that  God  or  man  willingly  makes  himself  in 
any  sense  worse  ? Impossible,  said  he.  Impossible,  then,  it  is, 
said  I,  that  a god  should  wish  to  change  himself;  but  ever  being 
fairest  and  best,  each  of  them  ever  remains  absolutely  in  the  same 
form.” 

The  premises  of  the  above  argument  are  the  following : 

1 st.  If  the  Deity  suffers  change,  He  is  changed  either  by  Him- 
self or  by  another. 

2nd.  If  He  is  in  the  best  state,  He  is  not  changed  by  another. 

3rd.  The  Deity  is  in  the  best  state. 

4th.  If  the  Deity  is  changed  by  Himself,  He  is  changed  to  a 
worse  state. 

5 th.  If  He  acts  willingly,  He  is  not  changed  to  a worse  state. 

6th.  The  Deity  acts  willingly. 

Let  us  express  the  elements  of  these  premises  as  follows : 

Let  x repi’esent,  the  proposition,  “ The  Deity  suffers  change.” 
g,  He  is  changed  by  Himself. 
z,  He  is  changed  by  another. 

s,  He  is  in  the  best  state. 

t,  He  is  changed  to  a worse  state. 
w,  He  acts  willingly. 

Then  the  premises  expressed  in  symbolical  language  yield,  after 
elimination  of  the  indefinite  class  symbols  v , the  following  equa- 
tions : 


CHAP.  XII.] 


METHODS  IN  SECONDARY  PROPOSITIONS. 


183 


xyz  + «(l  -y)  (1  - z)  = 0,  (1) 

sz  = 0,  (2) 

* = 1,  (3) 

y(l  - if)  = 0,  (4) 

wt  = 0,  (5) 

w = 1.  (6) 


Retaining  x,  I shall  eliminate  in  succession  z,  s,  y,  t,  and  to  (this 
being  the  order  in  which  those  symbols  occur  in  the  above  sys- 
tem), and  interpret  the  successive  results. 

Eliminating  z from  (1)  and  (2),  we  get 

xs(l-y)  = 0.  (7) 

Eliminating  s from  (3)  and  (7), 

*(1-30  = 0-  (8) 

Eliminating  y from  (4)  and  (8), 

*(1-0  = 0.  (9) 

Eliminating  t from  (5)  and  (9), 

xw  = 0.  (10) 

Eliminating  w from  (6)  and  (10), 

* = 0.  (11) 

These  equations,  beginning  with  (8),  give  the  following 
results : 

0 

From  (8)  we  have  x - - y,  therefore,  If  the  Deity  suffers 

change,  He  is  changed  by  Himself. 

0 

From  (9),  x = — t,  If  the  Deity  suffers  change,  He  is  changed 
to  a worse  state. 

From  (10),  x = ^ (1  - w ).  If  the  Deity  suffers  change,  He 
does  not  act  willingly. 

From  (11),  The  Deity  does  not  suffer  change.  This  is  Plato’s 
result. 

Now  I have  before  remarked,  that  the  order  of  elimination 
is  indifferent.  Let  us  in  the  present  case  seek  to  verify  this  fact 
by  eliminating  the  same  symbols  in  a reverse  order,  beginning 
with  to.  The  resulting  equations  are, 


184  METHODS  IN  SECONDARY  PROPOSITIONS.  [CHAP.  XII. 

t = o,  y = 0,  z ( 1 - z)  = 0,  z = 0,  x = 0 ; 

yielding  the  following  interpretations  : 

God  is  not  changed  to  a worse  state. 

He  is  not  changed  by  Himself. 

If  He  suffers  change.  He  is  changed  by  another. 

He  is  not  changed  by  another. 

He  is  not  changed. 

We  thus  reach  by  a different  route  the  same  conclusion. 

Though  as  an  exhibition  of  the  power  of  the  method,  the 
above  examples  are  of  slight  value,  they  serve  as  well  as  more 
complicated  instances  Avould  do,  to  illustrate  its  nature  and  cha- 
racter. 

7.  It  may  be  remarked,  as  affinal  instance  of  analogy  between 
the  system  of  primary  and  that  of  secondary  propositions,  that 
in  the  latter  system  also  the  fundamental  equation, 

x (1  - x)  = 0, 

admits  of  interpretation.  It  expresses  the  axiom,  A proposition 
cannot  at  the  same  time  be  true  and  false.  Let  this  be  compared 
with  the  corresponding  interpretation  (III.  15).  Solved  under 
the  form 

0 0 


by  development,  it  furnishes  the  respective  axioms  : “A  thing  is 
what  it  is:”  “ If  a proposition  is  true,  it  is  true forms  of  what  has 
been  termed  “ The  principle  of  identity.”  Upon  the  nature  and 
the  value  of  these  axioms  the  most  opposite  opinions  have  been 
entertained.  Some  have  regarded  them  as  the  very  pith  and  mar- 
row of  philosophy.  Locke  devoted  to  them  a chapter,  headed, 
“ On  Trifling  Propositions.”  * In  both  these  views  there  seems 
to  have  been  a mixture  of  truth  and  error.  Regarded  as  sup- 
planting experience,  or  as  furnishing  materials  for  the  vain  and 
wordy  janglings  of  the  schools,  such  propositions  are  worse  than 
trifling.  Viewed,  on  the  other  hand,  as  intimately  allied  with 
the  very  laws  and  conditions  of  thought,  they  rise  into  at  least  a 
speculative  importance. 


* Essay  on  the  Human  Understanding,  Book  IV.  Chap.  viii. 


CHAl’.  XIII.] 


CLARKE  AND  SPINOZA. 


185 


CHAPTER  XIII. 

ANALYSIS  OF  A PORTION  OF  DR.  SAMUEL  CLARKE’S  “DEMONSTRA- 
TION OF  THE  BEING  AND  ATTRIBUTES  OF  GOD,”  AND  OF  A 
PORTION  OF  THE  “ ETHICA  ORDINE  GEOMETRICO  DEMON- 
STRATA”  OF  SPINOZA. 

1 • HPHE  general  order  which,  in  the  investigations  of  the  fol- 
lowing  chapter,  I design  to  pursue,  is  the  following.  I 
shall  examine  what  are  the  actual  premises  involved  in  the  de- 
monstrations of  some  of  the  general  propositions  of  the  above 
treatises,  whether  those  premises  be  expressed  or  implied.  By 
the  actual  premises  I mean  whatever  propositions  are  assumed 
in  the  course  of  the  argument,  without  being  proved,  and  are 
employed  as  parts  of  the  foundation  upon  which  the  final  conclu- 
sion is  built.  The  premises  thus  determined,  I shall  express  in 
the  language  of  symbols,  and  I shall  then  deduce  from  them  by 
the  methods  developed  in  the  previous  chapters  of  this  work,  the 
most  important  inferences  which  they  involve,  in  addition  to  the 
particular  inferences  actually  drawn  by  the  authors.  I shall  in 
some  instances  modify  the  premises  by  the  omission  of  some  fact 
or  principle  which  is  contained  in  them,  or  by  the  addition  or 
substitution  of  some  new  proposition,  and  shall  determine  how 
by  such  change  the  ultimate  conclusions  are  affected.  In  the 
pursuit  of  these  objects  it  will  not  devolve  upon  me  to  inquire, 
except  incidentally,  how  far  the  metaphysical  principles  laid  down 
in  these  celebrated  productions  are  worthy  of  confidence,  but 
only  to  ascertain  what  conclusions  may  justly  be  drawn  from 
given  premises ; and  in  doing  this,  to  exemplify  the  perfect  li- 
berty which  we  possess  as  concerns  both  the  choice  and  the 
order  of  the  elements  of  the  final  or  concluding  propositions,  viz., 
as  to  determining  what  elementary  propositions  are  true  or  false, 
and  what  are  true  or  false  under  given  restrictions,  or  in  given 
combinations. 

2.  The  chief  practical  difficulty  of  this  inquiry  will  consist, 


186  CLARKE  AND  SPINOZA.  [CHAP.  XIII. 

not  in  the  application  of  the  method  to  the  premises  once  deter- 
mined, but  in  ascertaining  what  the  premises  are.  In  what  are 
regarded  as  the  most  rigorous  examples  of  reasoning  applied  to 
metaphysical  questions,  it  will  occasionally  be  found  that  different 
trains  of  thought  arc  blended  together;  that  particular  but  essen- 
tial parts  of  the  demonstration  are  given  parenthetically,  or  out 
of  the  main  course  of  the  argument;  that  the  meaning  of  a pre- 
miss may  be  in  some  degree  ambiguous ; and,  not  unfrequently, 
that  arguments,  viewed  by  the  strict  laws  of  formal  reasoning, 
are  incorrect  or  inconclusive.  The  difficulty  of  determining  and 
distinctly  exhibiting  the  true  premises  of  a demonstration  may, 
in  such  cases,  be  very  considerable.  But  it  is  a difficulty  which 
must  be  overcome  by  all  who  would  ascertain  whether  a parti- 
cular conclusion  is  proved  or  not,  whatever  form  they  may  be 
prepared  or  disposed  to  give  to  the  ulterior  process  of  reasoning. 
It  is  a difficulty,  therefore,  which  is  not  peculiar  to  the  method 
of  this  work,  though  it  manifests  itself  more  distinctly  in  con- 
nexion with  this  method  than  with  any  other.  So  intimate,  in- 
deed, is  this  connexion,  that  it  is  impossible,  employing  the  me- 
thod of  this  treatise,  to  form  even  a conjecture  as  to  the  validity 
of  a conclusion,  without  a distinct  apprehension  and  exact  state- 
ment of  all  the  premises  upon  which  it  rests.  In  the  more  usual 
course  of  procedure,  nothing  is,  however,  more  common  than  to 
examine  some  of  the  steps  of  a train  of  argument,  and  thence  to 
form  a vague  general  impression  of  the  scope  of  the  whole,  with- 
out any  such  preliminary  and  thorough  analysis  of  the  premises 
which  it  involves. 

The  necessity  of  a rigorous  determination  of  the  real  pre- 
mises of  a demonstration  ought  not  to  be  regarded  as  an  evil ; 
especially  as,  when  that  task  is  accomplished,  every  source  of 
doubt  or  ambiguity  is  removed.  In  employing  the  method  of 
this  treatise,  the  order  in  which  premises  are  arranged,  the  mode 
of  connexion  which  they  exhibit,  with  every  similar  circumstance, 
may  be  esteemed  a matter  of  indifference,  and  the  process  of 
inference  is  conducted  with  a precision  which  might  almost  be 
termed  mechanical. 

3.  The  “ Demonstration  of  the  Being  and  Attributes  of 
God,”  consists  of  a scries  of  propositions  or  theorems,  each 


CLARKE  AND  SPINOZA. 


18T 


CHAP.  XIII.] 

of  them  proved  by  means  of  premises  resolvable,  for  the  most 
part,  into  two  distinct  classes,  viz.,  facts  of  observation,  such 
as  the  existence  of  a material  world,  the  phamomenon  of  mo- 
tion, &c.,  and  hypothetical  principles,  the  authority  and  uni- 
versality of  which  are  supposed  to  be  recognised  a priori.  It  is, 
of  course,  upon  the  truth  of  the  latter,  assuming  the  correctness 
of  the  reasoning,  that  the  validity  of  the  demonstration  really  de- 
pends. But  whatever  may  be  thought  of  its  claims  in  this  re- 
spect, it  is  unquestionable  that,  as  an  intellectual  performance,  its 
merits  are  very  high.  Though  the  trains  of  argument  of  which 
it  consists  are  not  in  general  very  clearly  arranged,  they  are  al- 
most always  specimens  of  correct  Logic,  and  they  exhibit  a 
subtlety  of  apprehension  and  a force  of  reasoning  which  have 
seldom  been  equalled,  never  perhaps  surpassed.  We  see  in  them 
the  consummation  of  those  intellectual  efforts  which  were  awa- 
kened in  the  realm  of  metaphysical  inquiry,  at  a period  when  the 
dominion  of  hypothetical  principles  was  less  questioned  than  it 
now  is,  and  when  the  rigorous  demonstrations  of  the  newly  risen 
school  of  mathematical  physics  seemed  to  have  furnished  a model 
for  their  direction.  They  appear  to  me  for  this  reason  (not  to 
mention  the  dignity  of  the  subject  of  which  they  treat)  to  be 
deserving  of  high  consideration  ; and  I do  not  deem  it  a vain 
or  superfluous  task  to  expend  upon  some  of  them  a careful 
analysis. 

4.  The  Ethics  of  Benedict  Spinoza  is  a treatise,  the  object 
of  which  is  to  prove  the  identity  of  God  and  the  universe,  and 
to  establish,  upon  this  doctrine,  a system  of  morals  and  of  philo- 
sophy. The  analysis  of  its  main  argument  is  extremely  difficult, 
owing  not  to  the  complexity  of  the  separate  propositions  which  it 
involves,  but  to  the  use  of  vague  definitions,  and  of  axioms  which, 
through  a like  defect  of  clearness,  it  is  perplexing  to  determine 
whether  we  ought  to  acceptor  to  reject.  While  the  reasoning  of 
Dr.  Samuel  Clarke  is  in  part  verbal,  that  of  Spinoza  is  so  in  a much 
greater  degree ; and  perhaps  this  is  the  reason  why,  to  some 
minds,  it  has  appeared  to  possess  a formal  cogency,  to  which  in 
reality  it  possesses  no  just  claim.  These  points  will,  however, 
be  considered  in  the  proper  place. 


188 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 


clarke’s  demonstration. 

Proposition  I. 

5.  “ Something  has  existed  from  eternity .” 

The  proof  is  as  follows  : — 

“ For  since  something  now  is,  ’tis  manifest  that  something 
always  was.  Otherwise  the  things  that  now  are  must  have  risen 
out  of  nothing,  absolutely  and  without  cause.  Which  is  a 
plain  contradiction  in  terms.  For  to  say  a thing  is  produced, 
and  yet  that  there  is  no  cause  at  all  of  that  production,  is  to  say 
that  something  is  effected  when  it  is  effected  by  nothing,  that  is, 
at  the  same  time  when  it  is  not  effected  at  all.  Whatever  exists 
has  a cause  of  its  existence,  either  in  the  necessity  of  its  own 
nature,  and  thus  it  must  have  been  of  itself  eternal : or  in  the 
will  of  some  other  being,  and  then  that  other  being  must,  at  least 
in  the  order  of  nature  and  causality,  have  existed  before  it.” 

Let  us  now  proceed  to  analyze  the  above  demonstration.  Its 
first  sentence  is  resolvable  into  the  following  propositions : 

1st.  Something  is. 

2nd.  If  something  is,  either  something  always  was,  or  the 
things  that  now  are  must  have  risen  out  of  nothing. 

The  next  portion  of  the  demonstration  consists  of  a proof 
that  the  second  of  the  above  alternatives,  viz.,  “ The  things  that 
now  are  have  risen  out  of  nothing,”  is  impossible,  and  it  may 
formally  be  resolved  as  follows  : 

3rd.  If  the  things  that  now  are  have  risen  out  of  nothing, 
something  has  been  effected,  and  at  the  same  time  that  some- 
thing has  been  effected  by  nothing. 

4th.  If  that  something  has  been  effected  by  nothing,  it  has 
not  been  effected  at  all. 

The  second  portion  of  this  argument  appears  to  be  a mere 
assumption  of  the  point  to  be  proved,  or  an  attempt  to  make  that 
point  clearer  by  a different  verbal  statement. 

The  third  and  last  portion  of  the  demonstration  contains  a dis- 
tinct proof  of  the  truth  of  either  the  original  proposition  to  be 
proved,  viz.,  “ Something  always  was,”  or  the  point  proved  in 
the  second  part  of  the  demonstration,  viz.,  the  untenable  nature 


CLARKE  AND  SPINOZA. 


189 


CHAP.  XIII.] 

of  the  hypothesis,  that  “ the  things  that  now  are  have  risen  out 
of  nothing.”  It  is  resolvable  as  follows  : — 

5th.  If  something  is,  either  it  exists  by  the  necessity  of  its 
own  nature,  or  it  exists  by  the  will  of  another  being. 

6th.  If  it  exists  by  the  necessity  of  its  own  nature,  something 
always  was. 

7th.  If  it  exists  by  the  will  of  another  being,  then  the  pro- 
position, that  the  things  which  exist  have  arisen  out  of  nothing, 
is  false. 

The  last  proposition  is  not  expressed  in  the  same  form  in  the 
text  of  Dr.  Clarke ; but  his  expressed  conclusion  of  the  prior  ex- 
istence of  another  Being  is  clearly  meant  as  equivalent  to  a de- 
nial of  the  proposition  that  the  things  which  now  are  have  risen 
out  of  nothing. 

It  appears,  therefore,  that  the  demonstration  consists  of  two 
distinct  trains  of  argument : one  of  those  trains  comprising  what 
I have  designated  as  the  first  and  second  parts  of  the  demonstra- 
tion ; the  other  comprising  the  first  and  third  parts.  Let  us  con- 
sider the  latter  train. 

The  premises  are  : — 

1st.  Something  is. 

2nd.  If  something  is,  either  something  always  was,  or  the 
things  that  now  are  have  risen  out  of  nothing. 

3rd.  If  something  is,  either  it  exists  in  the  necessity  of  its 
own  nature,  or  it  exists  by  the  will  of  another  being. 

4 th.  If  it  exists  in  the  necessity  of  its  own  nature,  something 
always  was. 

5 th.  If  it  exists  by  the  will  of  another  being,  then  the  hy- 
pothesis, that  the  things  which  now  are  have  risen  out  of  nothing, 
is  false. 

We  must  now  express  symbolically  the  above  proposition. 

Let  x = Something  is. 

y = Something  always  was. 
z = The  things  which  now  are  have  risen  from 
nothing. 

jo  = It  exists  in  the  necessity  of  its  own  nature 
(i.  e.  the  something  spoken  of  above). 
q - It  exists  by  the  will  of  another  Being. 


190 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 


It  must  be  understood,  that  by  the  expression,  Let  x ~ 
“Something  is,”  is  meant  no  more  than  that  x is  the  repre-* 
sentative  symbol  of  that  proposition  (XI.  7),  the  equations 
x = 1,  x = 0,  respectively  declaring  its  truth  and  its  falsehood. 
The  equations  of  the  premises  are : — 

1st.  x = 1 ; 

2nd.  x = v [y  (1  - z)  + z (1  - ?/)}; 

3rd.  x = v\p(\- q)  + q(\-p))-, 

4th.  p = vy; 

5th.  q = v (1  - z)’, 

and  on  eliminating  the  several  indefinite  symbols  v,  we  have 


1_^  = 0;  (1) 

x[yz+(l-y)(\-z)}  = 0-,  (2) 

x \V<1  + (i  -/>)(i  - ?)}  = 0;  (3) 

p(l~y)  = o»  (4) 

qz  = 0.  (5) 


6.  First,  I shall  examine  whether  any  conclusions  are  dedu- 
cible  from  the  above,  concerning  the  truth  or  falsity  of  the 
single  propositions  represented  by  the  symbols  y,  z,  p,  q,  viz.,  of 
the  propositions,  “ Something  always  was  “ The  things  which 
now  are  have  risen  from  nothing;”  “The  something  which  is 
exists  by  the  necessity  of  its  own  nature “ The  something 
which  is  exists  by  the  will  of  another  being.” 

For  this  purpose  we  must  separately  eliminate  all  the  symbols 
but  y,  all  these  but  z,  &c.  The  resulting  equation  will  deter- 
mine whether  any  such  separate  relations  exist. 

To  eliminate  x from  (1),  (2),  and  (3),  it  is  only  necessary  to 
substitute  in  (2)  and  (3)  the  value  of  x derived  from  (1).  We 
find  as  the  results, 

yz  + (i  -y)  0 - z)  = °-  (6) 

Pi  + (i-p)(i-q)  = o.  (7) 

To  eliminate  p we  have  from  (4)  and  (7),  by  addition, 

pO  - y)  +pq  + 0-t)  (1-?)  = °;  (8) 

whence  we  find, 


(i  -y)  (1  " l)  = °- 


(9) 


CHAP.  XIII.]  CLARKE  AND  SPINOZA.  191 

To  eliminate  q from  (5)  and  (9),  we  have 
whence  we  find 

2(1-3/)  = 0.  (10) 

There  now  remain  but  the  two  equations  (6)  and  (10),  which, 
on  addition,  give 

yz  + l - y = 0. 

Eliminating  from  this  equation  z,  we  have 

1 - V = 0,  or,  y = 1.  (11) 

Eliminating  from  the  same  equation  y,  we  have 

z = 0.  (12) 

The  interpretation  of  (11)  is 

Something  always  was. 

The  interpretation  of  (12)  is 

The  things  which  are  have  not  risen  from  nothing. 

Next  resuming  the  system  (6),  (7),  with  the  two  equations 
(4),  (5),  let  us  determine  the  two  equations  involving  p and  q 
respectively. 

To  eliminate  y we  have  from  (4)  and  (6), 

p (i  - y)  + yz  + (i  - y)  0 - *)  = (o) ; 

whence  (p  + 1 - 2)  2 = 0,  or,  pz  = 0.  (13) 

To  eliminate  2 from  (5)  and  (13),  we  have 
qz  + pz  = 0 ; 

whence  we  get, 

0 = 0. 

There  remains  then  but  the  equation  (7),  from  which  elimi- 
nating q,  we  have  0 = 0 for  the  final  equation,  in  p. 

Hence  there  is  no  conclusion  derivable  from  the  premises  af- 
firming the  simple  truth  or  falsehood  of  the  proposition , “ The, 
something  which  is  exists  in  the  necessity  of  its  own  nature And  as, 
on  eliminating  p,  there  is  the  same  result,  0 = 0,  for  the  ultimate 
equation  in  q,  it  also  follows,  that  there  is  no  conclusion  deducihle 
from  the  premises  as  to  the  simple  truth  or  falsehood  of  the  propo- 
sition, “ The  something  which  is  exists  Inj  the  will  of  another  Being.” 


192 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 

Of  relations  connecting  more  than  one  of  the  propositions  re- 
presented by  the  elementary  symbols,  it  is  needless  to  consider 
any  but  that  which  is  denoted  by  the  equation  (7)  connecting 
p and  q,  inasmuch  as  the  propositions  represented  by  the  remain- 
ing symbols  are  absolutely  true  or  false  independently  of  any  con- 
nexion of  the  kind  here  spoken  of.  The  interpretation  of  (7), 
placed  under  the  form 

P(l~Q)  + ?(1  ~P)  = 1>  ls> 

The  something  which  is,  either  exists  in  the  necessity  of  its 
own  nature,  or  by  the  viill  of  another  being. 

I have  exhibited  the  details  of  the  above  analysis  with  a, 
perhaps,  needless  fulness  and  prolixity,  because  in  the  examples 
which  will  follow,  I propose  rather  to  indicate  the  steps  by 
which  results  are  obtained,  than  to  incur  the  danger  of  a weari- 
some frequency  of  repetition.  The  conclusions  which  have  re- 
sulted from  the  above  application  of  the  method  are  easily  verified 
by  ordinary  reasoning. 

The  reader  will  have  no  difficulty  in  applying  the  method 
to  the  other  train  of  premises  involved  in  Dr.  Clarke’s  first  Pro- 
position, and  deducing  from  them  the  two  first  of  the  conclusions 
to  which  the  above  analysis  has  led. 

Proposition  II. 

7.  Some  one  unchangeable  and  independent  Being  has  existed 
from  eternity. 

The  premises  from  which  the  above  proposition  is  proved 
are  the  following : 

1st.  Something  has  always  existed. 

2nd.  If  something  has  always  existed,  either  there  has  existed 
some  one  unchangeable  and  independent  being,  or  the  whole  of 
existing  things  has  been  comprehended  in  a succession  of  change- 
able and  dependent  beings. 

3rd.  If  the  universe  has  consisted  of  a succession  of  change- 
able and  dependent  beings,  either  that  series  has  had  a cause  from 
without,  or  it  has  had  a cause  from  within. 

4th.  It  has  not  had  a cause  from  without  (because  it  includes, 
by  hypothesis,  all  things  that  exist). 


CLARKE  AND  SPINOZA. 


193 


CHAP.  XIII.] 


5 th.  It  has  not  had  a cause  from  within  (because  no  part  is 
necessary,  and  if  no  part  is  necessary,  the  whole  cannot  be  ne- 
cessary). 

Omitting,  merely  for  brevity,  the  subsidiary  proofs  contained 
in  the  parentheses  of  the  fourth  and  fifth  premiss,  we  may  repre- 
sent the  premises  as  follows : 

Let  x = Something  has  always  existed. 

y = There  has  existed  some  one  unchangeable  and  in- 
dependent being. 

z = There  has  existed  a succession  of  changeable  and 
dependent  beings. 

p = That  series  has  had  a cause  from  without. 
q = That  series  has  had  a cause  from  within. 

Then  we  have  the  following  system  of  equations,  viz. : 

1st.  x = 1 ; 

2nd.  x = v [y  (\  - z)  + z(\  - y))  ; 

3rd.  z = v [p  (1  - q)  + (1  - p)  q] ; 

4th.  p = 0 ; 

5th.  q '=  0 : 

which,  on  the  separate  elimination  of  the  indefinite  symbols  v. 


gives 

1 - a;  = 0 ; (1) 

*{y*  + (i-y)(i-*)}  = °;  (2) 

z lpq+  C1  -p)  (!  - ?)}  = 0 ; (3) 

P = 0 ; (4) 

q = 0.  (5) 


The  elimination  from  the  above  system  of  x,  p,  q,  and  y,  con- 
ducts to  the  equation 

z = 0. 


And  the  elimination  ofz,  p,  y,  and  z,  conducts  in  a similar  man- 
ner to  the  equation 

y = i- 


Of  which  equations  the  respective  interpretations  are  : 

1st.  The  whole  of  existing  things  has  not  been  comprehended 
in  a succession  of  changeable  and  dependent  beings. 

2nd.  There  has  existed  some  one  unchangeable  and  independent 
being. 


194 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 

The  latter  of  these  is  the  proposition  which  Dr.  Clarke  proves. 
As,  by  the  above  analysis,  all  the  propositions  represented  by  the 
literal  symbols  x,  y,  z,  p,  q,  are  determined  as  absolutely  true  or 
false,  it  is  needless  to  inquire  into  the  existence  of  any  further  re- 
lations connecting  those  propositions  together. 

Another  proof  is  given  of  Prop,  n.,  which  for  brevity  I pass 
over.  It  may  be  observed,  that  the  “ impossibility  of  infinite 
succession,”  the  proof  of  which  forms  a part  of  Clarke’s  argu- 
ment, has  commonly  been  assumed  as  a fundamental  principle  of 
metaphysics,  and  extended  to  other  questions  than  that  of  causa- 
tion. Aristotle  applies  it  to  establish  the  necessity  of  first  prin- 
ciples of  demonstration  ;*  the  necessity  of  an  end  (the  good),  in 
human  actions,  &c.f  There  is,  perhaps,  no  principle  more  fre- 
quently referred  to  in  his  writings.  By  the  schoolmen  it  was 
similarly  applied  to  prove  the  impossibility  of  an  infinite  subor- 
dination of  genera  and  species,  and  hence  the  necessary  existence 
of  universals.  Apparently  the  impossibility  of  our  forming  a 
definite  and  complete  conception  of  an  infinite  series,  i.  e.  of 
comprehending  it  as  a whole,  has  been  confounded  with  a logical 
inconsistency,  or  contradiction  in  the  idea  itself. 

8.  The  analysis  of  the  following  argument  depends  upon  the 
theory  of  Primary  Propositions. 


Proposition  III. 

That  unchangeable  and  independent  Being  must  he  self-existent. 

The  premises  are  : — 

1.  Every  being  must  either  have  come  into  existence  out  of 
nothing,  or  it  must  have  been  produced  by  some  external  cause, 
or  it  must  be  self-existent. 

2.  No  being  has  come  into  existence  out  of  nothing. 

3.  The  unchangeable  and  independent  Being  has  not  been 
produced  by  an  external  cause. 

For  the  symbolical  expression  of  the  above,  let  us  assume, 


* Metaphysics,  III.  4 ; Anal.  Post.  I.  li),  ct  se <j. 
f Nic.  Ethics,  Book  I.  Cap.  n. 


CHAP.  XIII.] 


CLARKE  AND  SPINOZA. 


195 


x = Beings  which  have  arisen  out  of  nothing. 
y = Beings  which  have  been  produced  by  an  external 
cause. 

z = Beings  which  are  self-existent. 
w = The  unchangeable  and  independent  Being. 

Then  we  have 

*0  - y){ 1 ~z)  + y(}  -*)(l  ~z)  + z{\-x)  (1  -y)  = 1,  (1) 

x = 0,  (2) 

w = v(  l-y),  (3) 

from  the  last  of  which  eliminating  v, 

wy  = 0.  (4) 

Whenever,  as  above,  the  value  of  a symbol  is  given  as  0 or  1,  it 
is  best  eliminated  by  simple  substitution.  Thus  the  elimination 
of  x gives 

y 0 - z)  + *0  -y)  = 1 5 (5) 

or,  yz  + (1  - y)  (1  - z)  = 0.  (6) 

Now  adding  (4)  and  (6),  and  eliminating  y,  we  get 

w ( 1 - z)  = 0, 
iv  = vz ; 

the  interpretation  of  which  is, — The  unchangeable  and  indepen- 
dent being  is  necessarily  self-existing . 

Of  (5),  in  its  actual  form,  the  interpretation  is, — Every  being 
has  either  been  produced  by  an  external  cause , or  it  is  self-existent. 

9.  In  Dr.  Samuel  Clarke’s  observations  on  the  above  propo- 
sition occurs  a remarkable  argument,  designed  to  prove  that  the 
material  world  is  not  the  self-existent  being  above  spoken  of. 
The  passage  to  which  I refer  is  the  following  : 

“ If  matter  be  supposed  to  exist  necessarily,  then  in  that  ne- 
cessary existence  there  is  either  included  the  power  of'  gravitation, 
or  not.  If  not,  then  in  a world  merely  material,  and  in  which  no 
intelligent  being  presides,  there  never  could  have  been  any  mo- 
tion ; because  motion,  as  has  been  already  shown,  and  is  now 
granted  in  the  question,  is  not  necessary  of  itself.  But  if  the 


19G 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 

power  of  gravitation  be  included  in  the  pretended  necessary  ex- 
istence of  matter : then,  it  following  necessarily  that  there  must 
be  a vacuum  (as  the  incomparable  Sir  Isaac  Newton  has  abun- 
dantly demonstrated  that  there  must,  if  gravitation  be  an  uni- 
versal quality  or  affection  of  matter),  it  follows  likewise,  that 
matter  is  not  a necessary  being.  For  if  a vacuum  actually  be, 
then  it  is  plainly  more  than  possible  for  matter  not  to  be.” — 
(pp.  25,  2G). 

It  will,  upon  attentive  examination,  be  found  that  the  actual 
premises  involved  in  the  above  demonstration  are  the  following  : 

1st.  If  matter  is  a necessary  being,  either  the  property  of  gra- 
vitation is  necessarily  present,  or  it  is  necessarily  absent. 

2nd.  If  gravitation  is  necessarily  absent,  and  the  world  is  not 
subject  to  any  presiding  Intelligence,  motion  does  not  exist. 

3rd.  If  the  property  of  gravitation  is  necessarily  present,  the 
existence  of  a vacuum  is  necessary. 

4th.  If  the  existence  of  a vacuum  is  necessary,  matter  is  not  a 
necessary  being. 

5th.  If  matter  is  a necessary  being,  the  world  is  not  subject 
to  a presiding  Intelligence. 

6th.  Motion  exists. 

Of  the  above  premises  the  first  four  are  expressed  in  the  de- 
monstration ; the  fifth  is  implied  in  the  connexion  of  its  first  and 
second  sentences ; and  the  sixth  expresses  a fact,  which  the  au- 
thor does  not  appear  to  have  thought  it  necessary  to  state,  but 
which  is  obviously  a part  of  the  ground  of  his  reasoning.  Let  us 
represent  the  elementary  propositions  in  the  following  manner : 

Let  x = Matter  is  a necessary  being. 

y = Gravitation  is  necessarily  present. 
t = Gravitation  is  necessarily  absent. 

2 = The  world  is  merely  material,  and  not  subject  to 
any  presiding  Intelligence. 
w = Motion  exists. 
v = A vacuum  is  necessary. 

Then  the  system  of  premises  will  be  represented  by  the  following 
equations,  in  which  <y  is  employed  as  the  symbol  ot  time  indefi- 
nite : 


CHAI\  XIII.] 


CLARKE  AND  SPINOZA. 


197 


x = <1  (y(!  - 0 + 0 -y)*r 

tz  = q (1  - iv). 

y = qv- 

v = q (1  - xf 
x = qz. 
iv  = 1. 

From  which,  if  we  eliminate  the  symbols  q,  we  have  the  follow- 


ing system,  viz. : 

* [yt+  (i  -y)  (i  - 0)  = (0 

tzw  = 0.  (2) 

y(  i-»)  = o.  (3) 

vx  = 0.  (4) 

a-  (1  - 2)  = 0.  (5) 

1 - iv  = 0.  (6) 


Now  if  from  these  equations  we  eliminate  iv,  v,  z,  y,  and  t,  we 
obtain  the  equation 

x = 0, 

which  expresses  the  proposition,  Matter  is  not  a necessary  being. 
This  is  Dr.  Clarke’s  conclusion.  If  we  endeavour  to  eliminate 
any  other  set  of  five  symbols  (except  the  set  v,  z,  y,  t,  and  x, 
which  would  give  iv  - 1),  Ave  obtain  a result  of  the  form  0 = 0. 
It  hence  appears  that  there  are  no  other  conclusions  expressive  of 
the  absolute  truth  or  falsehood  of  any  of  the  elementary  propositions 
designated  by  single  symbols. 

Of  conclusions  expressed  by  equations  involving  tAVO  symbols, 
there  exists  but  the  following,  viz. : — If  the  world  is  merely  mate- 
rial and  not  subject  to  a presiding  Intelligence , gravitation  is  not 
necessarily  absent.  This  conclusion  is  expressed  by  the  equation 

tz  = 0,  whence  z = q (1  - t). 

If  in  the  above  analysis  we  suppress  the  concluding  premiss,  ex- 
pressing the  fact  of  the  existence  of  motion,  and  leave  the  hypo- 
thetical principles  Avhich  are  embodied  in  the  remaining  premises 
untouched,  some  remarkable  conclusions  follow.  To  these  I 
shall  direct  attention  in  the  folloAving  chapter. 

10.  Of  the  remainder  of  Dr.  Clarke’s  argument  I shall  briefly 
state  the  substance  and  connexion,  dAvelling  only  on  certain  por- 


198 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 

tions  of  it  which  are  of  a more  complex  character  than  the  others, 
and  afford  better  illustrations  of  the  method  of  this  work. 

In  Prop.  iv.  it  is  shown  that  the  substance  or  essence  of  the 
self- existent  being  is  incomprehensible.  The  tenor  of  the  reason- 
ing employed  is,  that  we  are  ignorant  of  the  essential  nature  of 
all  other  things, — much  more,  then,  of  the  essence  of  the  self- 
existent  being. 

In  Prop.  v.  it  is  contended  that  “though  the  substance  or 
essence  of  the  self-existent  being  is  itself  absolutely  incompre- 
hensible to  us,  yet  many  of  the  essential  attributes  of  his  nature 
are  strictly  demonstrable,  as  well  as  his  existence.” 

In  Prop.  vi.  it  is  argued  that  “the  self-existent  being  must 
of  necessity  be  infinite  and  omnipresent and  it  is  contended 
that  his  infinity  must  be  “an  infinity  of  fulness  as  well  as  of 
immensity.”  The  ground  upon  which  the  demonstration  pro- 
ceeds is,  that  an  absolute  necessity  of  existence  must  be  inde- 
pendent of  time,  place,  and  circumstance,  free  from  limitation, 
and  therefore  excluding  all  imperfection.  And  hence  it  is  in- 
ferred that  the  self-existent  being  must  be  “ a most  simple,  un- 
changeable, incorruptible  being,  without  parts,  figure,  motion, 
or  any  other  such  properties  as  we  find  in  matter.” 

The  premises  actually  employed  may  be  exhibited  as  follows  : 

1.  If  a finite  being  is  self-existent,  it  is  a contradiction  to 
suppose  it  not  to  exist. 

2.  A finite  being  may,  without  contradiction,  be  absent  from 
one  place. 

3.  That  which  may  without  contradiction  be  absent  from  one 
place  may  without  contradiction  be  absent  from  all  places. 

4.  That  which  may  without  contradiction  be  absent  from  all 
places  may  without  contradiction  be  supposed  not  to  exist. 

Let  us  assume 
x - Finite  beings. 
y = Things  self-existent. 

2 = Things  which  it  is  a contradiction  to  suppose  not  to  exist. 
w = Things  which  may  be  absent  without  contradiction  from 
one  place. 

t = Things  which  without  contradiction  may  be  absent  from 
every  place. 


CHAP.  XIII.]  CLARKE  AND  SPINOZA.  199 

We  have  on  expressing  the  above,  and  eliminating  the  indefinite 
symbols, 


*y  (i  - 2)  = o. 

(1) 

x (1  - io)  =0. 

(2) 

w (1  - t)  =0. 

(3) 

tz  = 0. 

(4) 

Eliminating  in  succession  t,  iv,  and  z,  we  get 

xy  = 0, 

= \ 0 -*); 

the  interpretation  of  which  is, — Whatever  is  self-existent  is  in- 
finite. 

In  Prop.  vii.  it  is  argued  that  the  self-existent  being  must  of 
necessity  be  One.  The  order  of  the  proof  is,  that  the  self-exis- 
tent being  is  “necessarily  existent,”  that  “necessity  absolute  in 
itself  is  simple  and  uniform,  and  without  any  possible  difference 
or  variety,”  that  all  “ variety  or  difference  of  existence”  implies 
dependence ; and  hence  that  “ whatever  exists  necessarily  is  the 
one  simple  essence  of  the  self-existent  being.” 

The  conclusion  is  also  made  to  flow  from  the  following  pre- 
mises : — 

1 . If  there  are  two  or  more  necessary  and  independent  beings, 
either  of  them  may  be  supposed  to  exist  alone. 

2.  If  either  may  be  supposed  to  exist  alone,  it  is  not  a contra- 
diction to  suppose  the  other  not  to  exist. 

3.  If  it  is  not  a contradiction  to  suppose  this,  there  are  not 
two  necessary  and  independent  beings. 

Let  us  represent  the  elementary  propositions  as  follows  : — 
x = there  exist  two  necessary  independent  beings. 
y = either  may  be  supposed  to  exist  alone. 
z = it  is  not  a contradiction  to  suppose  the  other  not  to  exist. 


We  have  then,  on  proceeding  as  before, 

© 

ll 

l 

H 

(0 

2/(1  - z)  = 0. 

(2) 

© 

II 

M 

(3) 

‘200 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 


Eliminating  y and  z,  we  have 

x = 0. 

Whence,  There  do  not  exist  two  necessary  and  independent  beings. 

1 1 . To  the  premises  upon  which  the  two  previous  propositions 
rest,  it  is  well  known  that  Bishop  Butler,  who  at  the  time  of  the 
publication  of  the  “ Demonstration,”  was  a student  in  a non- 
conformist academy,  made  objection  in  some  celebrated  letters, 
which,  together  with  Dr.  Clarke’s  replies  to  them,  are  usually 
^appended  to  editions  of  the  work.  The  real  question  at  issue  is 
the  validity  of  the  principle,  that  “ whatsoever  is  absolutely  ne- 
•cessary  at  all  is  absolutely  necessary  in  every  part  of  space,  and 
in  every  point  of  duration,” — a principle  assumed  in  Dr.  Clarke’s 
reasoning,  and  explicitly  stated  in  his  reply  to  Butler’s  first  let- 
ter. In  his  second  communication  Butler  says : “ I do  not  con- 
ceive that  the  idea  of  ubiquity  is  contained  in  the  idea  of  self- 
existence, or  directly  follows  from  it,  any  otherwise  than  as  what- 
ever exists  must  exist  someichere.”  That  is  to  say,  necessary 
existence  implies  existence  in  some  part  of  space,  but  not  in 
every  part.  It  does  not  appear  that  Dr.  Clarke  was  ever  able  to 
dispose  effectually  of  this  objection.  The  whole  of  the  corres- 
pondence is  extremely  curious  and  interesting.  The  objections 
of  Butler  are  precisely  those  which  would  occur  to  an  acute  mind 
impressed  with  the  conviction,  that  upon  the  sifting  of  first  prin- 
ciples, rather  than  upon  any  mechanical  dexterity  of  reasoning, 
the  successful  investigation  of  truth  mainly  depends.  And  the 
replies  of  Dr.  Clarke,  although  they  cannot  be  admitted  as  satis- 
factory, evince,  in  a remarkable  degree,  that  peculiar  intellectual 
power  which  is  manifest  in  the  work  from  which  the  discussion 
arose. 

12.  In  Prop.  viii.  it  is  argued  that  the  self-existent  and  ori- 
ginal cause  of  all  things  must  be  an  Intelligent  Being. 

The  main  argument  adduced  in  support  of  this  proposition  is, 
that  as  the  cause  is  more  excellent  than  the  effect,  the  self- 
existent  being,  as  the  cause  and  original  of  all  things,  must  con- 
tain in  itself  the  perfections  of  all  things  ; and  that  Intelligence 
is  one  of  the  perfections  manifested  in  a part  of  the  creation.  It 
is  further  argued  that  this  perfection  is  not  a modification  of 


CHAP.  XIII.]  CLARKE  AND  SPINOZA.  201 

figure,  divisibility,  or  any  of  the  known  properties  of  matter ; 
for  these  are  not  perfections,  but  limitations.  To  this  is  added 
the  a posteriori  argument  from  the  manifestation  of  design  in  the 
frame  of  the  universe. 

There  is  appended,  however,  a distinct  argument  for  the 
existence  of  an  intelligent  self-existent  being,  founded  upon  the 
phenomenal  existence  of  motion  in  the  universe.  I shall  briefly 
exhibit  this  proof,  and  shall  apply  to  it  the  method  of  the  present 
treatise. 

The  argument,  omitting  unimportant  explanations,  is  as  fol- 
lows : — 

“ ’Tis  evident  there  is  some  such  a thing  as  motion  in  the 
world ; which  either  began  at  some  time  or  other,  or  was  eternal. 
If  it  began  in  time,  then  the  question  is  granted  that  the  first 
cause  is  an  intelligent  being.  . . . On  the  contrary,  if  motion  was 
eternal,  either  it  was  eternally  caused  by  some  eternal  intelligent 
being,  or  it  must  of  itself  be  necessary  and  self-existent,  or  else, 
without  any  necessity  in  its  own  nature,  and  without  any  external 
necessary  cause,  it  must  have  existed  from  eternity  by  an  endless 
successive  communication.  If  motion  was  eternally  caused  by 
some  eternal  intelligent  being,  this  also  is  granting  the  question 
as  to  the  present  dispute.  If  it  was  of  itself  necessary  and  self- 
existent,  then  it  follows  that  it  must  be  a contradiction  in  terms 
to  suppose  any  matter  to  be  at  rest.  And  yet,  at  the  same  time, 
because  the  determination  of  this  self-existent  motion  must  be 
every  way  at  once,  the  effect  of  it  would  be  nothing  else  but  a 
perpetual  rest.  . . . But  if  it  be  said  that  motion,  without  any  ne- 
cessity in  its  own  nature,  and  without  any  external  necessary 
cause,  has  existed  from  eternity  merely  by  an  endless  successive 
communication,  as  Spinoza  inconsistently  enough  seems  to  assert, 
this  I have  before  shown  (in  the  proof  of  the  second  general 
proposition  of  this  discourse)  to  be  a plain  contradiction.  It  re- 
mains, therefore,  that  motion  must  of  necessity  be  originally 
caused  by  something  that  is  intelligent.” 

The  premises  of  the  above  argument  may  be  thus  disposed  : 

1.  If  motion  began  in  time,  the  first  cause  is  an  intelligent 
being. 


202  CLARKE  AND  SPINOZA.  [CHAP.  XIII. 

2.  If  motion  has  existed  from  eternity,  either  it  has  been 
eternally  caused  by  some  eternal  intelligent  being,  or  it  is  self- 
existent,  or  it  must  have  existed  by  endless  successive  communi- 
cation. 

3.  If  motion  has  been  eternally  caused  by  an  eternal  intelli- 
gent being,  the  first  cause  is  an  intelligent  being. 

4 . If  it  is  self-existent,  matter  is  at  rest  and  not  at  rest. 

5.  That  motion  has  existed  by  endless  successive  communi- 
cation, and  that  at  the  same  time  it  is  not  self-existent,  and  has 
not  been  eternally  caused  by  some  eternal  intelligent  being,  is 
false. 

To  express  these  propositions,  let  us  assume — 
x = Motion  began  in  time  (and  therefore) 

1 - x = Motion  has  existed  from  eternity. 
y - The  first  cause  is  an  intelligent  being. 
p = Motion  has  been  eternally  caused  by  some  eternal  intelli- 
gent being. 

q = Motion  is  self-existent. 

r = Motion  has  existed  by  endless  successive  communication. 
s = Matter  is  at  rest. 

The  equations  of  the  premises  then  are — 
x — vy. 

i -x  = v {Ki-?)0  ~r)  + <?(l  ~P)  0 ~r)  + r(!  -p)0  -?)}• 

p = vy- 

q = vs  (1  - s')  - 0. 

r(l  ~ 9)  0 ~P)  = °- 

Since,  by  the  fourth  equation,  <7  = 0,  we  obtain,  on  substituting 
for  q its  value  in  the  remaining  equations,  the  system 

x = vy,  1 - x = v {/>  (1  - r)  + r (1  -p)), 
p = vy,  r ( 1 - p)  = 0, 

from  which  eliminating  the  indefinite  symbols  v,  we  have  the 
final  reduced  system, 


o' 

II 

1 

(1) 

(1  -x)  {pr  + (l  -/>)(!  -r)}  =0, 

(2) 

p(i-  y)  = 0. 

(3) 

r (1  - p)  = 0. 

(4) 

CLARKE  AND  SPINOZA. 


203 


CHAP.  XIII.] 

We  shall  first  seek  the  value  of  y,  the  symbol  involved  in  Dr. 
Clarke’s  conclusion.  First,  eliminating  x from  (1)  and  (2),  we 
have 

(1  -y)  [Pr+  0 -7>)0  ~r))  =0-  (5) 

Next,  to  eliminate  r from  (4)  and  (5),  we  have 

-p)  + 0 -y)  \pr  + O -p) 0 ~r)\  = °> 

•••  (1  -p  + 0 -y)p)  * C1  -y)  (i  -p)  = 0; 

whence 

(1  -y)  0 ~P)  = 0.  (6) 

Lastly,  eliminating  p from  (3)  and  (6),  we  have  - 
i -y  = o, 

•••  y = L 

which  expresses  the  required  conclusion,  The  first  cause  is  an 
intelligent  being. 

Let  us  now  examine  what  other  conclusions  are  deducible 
from  the  premises. 

If  we  substitute  the  value  just  found  for  y in  the  equations 
(1),  (2),  {3),  (4),  they  are  reduced  to  the  following  pair  of  equa- 
tions, viz., 

(1  - x)  (pr  + (1  -p)  (1  -r)}  = 0,  r(l-p)  = 0.  (7) 

Eliminating  from  these  equations  x,  we  have 
r (1  - p)  = 0,  whence  r - vp, 

which  expresses  the  conclusion,  If  motion  has  existed  by  endless 
successive  communication,  it  has  been  eternally  caused  by  an  eter- 
nal intelligent  being. 

Again  eliminating,  from  the  given  pair,  r,  we  have 
(1  - x)  (1  -p)  = 0, 
or,  1 - x - vp, 

which  expresses  the  conclusion,  If  motion  has  existed  from  eter- 
nity, it  has  been  eternally  caused  by  some  eternal  intelligent  being. 
Lastly,  from  the  same  original  pair  eliminating  p,  we  get 
(1  - x)  r = 0, 

which,  solved  in  the  form 

1 _ * = v (1  - r). 


CLARKE  AND  SPINOZA. 


204 


[chap.  XIII. 


gives  the  conclusion,  If  motion  has  existed  from  eternity , it  has  not 
existed  by  an  endless  successive  communication. 

Solved  under  the  form 

r = vx, 

the  above  equation  leads  to  the  equivalent  conclusion,  If  motion 
exists  by  an  endless  successive  communication , it  bey  an  in  time. 

13.  Now  it  will  appear  to  the  reader  that  the  first  and  last  of 
the  above  four  conclusions  are  inconsistent  with  each  other.  The 
two  consequences  drawn  from  the  hypothesis  that  motion  exists 
by  an  endless  successive  communication,  viz.,  1st,  that  it  has 
been  eternally  caused  by  an  eternal  intelligent  being ; 2ndly,  that 
it  began  in  time, — are  plainly  at  variance.  Nevertheless,  they  are 
both  rigorous  deductions  from  the  original  premises.  The  oppo- 
sition between  them  is  not  of  a logical , but  of  what  is  technically 
termed  a material  character.  This  opposition  might,  however, 
have  been  formally  stated  in  the  premises.  We  might  have 
added  to  them  a formal  proposition,  asserting  that  “ whatever  is 
eternally  caused  by  an  eternal  intelligent  being,  does  not  begin  in 
time.”  Had  this  been  done,  no  such  opposition  as  now  appears 
in  our  conclusions  could  have  presented  itself.  Formal  logic 
can  only  take  account  of  relations  which  are  formally  expressed 
(VI.  16);  and  it  may  thus,  in  particular  instances,  become  ne- 
cessary to  express,  in  a formal  manner,  some  connexion  among 
the  premises  which,  without  actual  statement,  is  involved  in  the 
very  meaning  of  the  language  employed. 

To  illustrate  what  has  been  said,  let  us  add  to  the  equations 
(2)  and  (4)  the  equation 

px  = 0, 

which  expresses  the  condition  above  adverted  to.  We  have 

(1  - x ) [pr  + (1  - p)  (1  - ?•)}  + r (I  - p)  + px  = 0.  (8) 

Eliminating  p from  this,  we  find,  simply 

r = 0, 

which  expresses  the  proposition,  Motion  does  not  exist  by  an  end- 
less successive  communication.  If  now  we  substitute  for  r its  value 
in  (8),  we  have 

(1  - x)  (1  - p)  l px  = 0,  or,  1 - x = p; 


CHAP.  XIII.] 


CLARKE  AND  SPINOZA. 


205 


whence  we  have  the  interpretation,  If  motion  has  existed  from 
eternity , it  has  been  eternally  caused  by  an  eternal  intelligent  being  ; 
together  with  the  converse  of  that  proposition. 

In  Prop.  ix.  it  is  argued,  that  “ the  self-existent  and  original 
cause  of  all  things  is  not  a necessary  agent,  but  a being  endued 
with  liberty  and  choice.”  The  proof  is  based  mainly  upon  his 
possession  of  intelligence,  and  upon  the  existence  of  final  causes, 
implying  design  and  choice.  To  the  objection  that  the  supreme 
cause  operates  by  necessity  for  the  production  of  what  is  best,  it 
is  replied,  that  this  is  a necessity  of  fitness  and  wisdom,  and  not 
of  nature. 

14.  In  Prop.  x.  it  is  argued,  that  “the  self-existent  being, 
the  supreme  cause  of  all  things,  must  of  necessity  have  infinite 
power.”  The  ground  of  the  demonstration  is,  that  as  “ all  the 
powers  of  all  things  are  derived  from  him,  nothing  can  make  any 
difficulty  or  resistance  to  the  execution  of  his  will.”  It  is  de- 
fined that  the  infinite  power  of  the  self-existent  being  does  not 
extend  to  the  “ making  of  a thing  which  implies  a contradiction,” 
or  the  doing  of  that  “ which  would  imply  imperfection  (whether 
natural  or  moral)  in  the  being  to  whom  such  power  is  ascribed,” 
but  that  it  does  extend  to  the  creation  of  matter,  and  of  an  im- 
material, cogitative  substance,  endued  with  a power  of  beginning 
motion,  and  with  a liberty  of  will  or  choice.  Upon  this  doctrine 
of  liberty  it  is  contended  that  we  are  able  to  give  a satisfactory 
answer  to  “that  ancient  and  great  question,  ttoObv  to  kokov, 
what  is  the  cause  and  original  of  evil  ?”  The  argument  on  this 
head  I shall  briefly  exhibit. 

“ All  that  we  call  evil  is  either  an  evil  of  imperfection,  as  the 
want  of  certain  faculties  or  excellencies  which  other  creatures 
have ; or  natural  evil,  as  pain,  death,  and  the  like  ; or  moral  evil, 
as  all  kinds  of  vice.  The  first  of  these  is  not  properly  an  evil ; 
for  every  power,  faculty,  or  perfection,  which  any  creature  enjoys, 
being  the  free  gift  of  God,  . . it  is  plain  the  want  of  any  certain 
faculty  or  perfection  in  any  kind  of  creatures,  which  never  be- 
longed to  their  natures  is  no  more  an  evil  to  them,  than  their 
never  having  been  created  or  brought  into  being  at  all  could  pro- 
perly have  been  called  an  evil.  The  second  kind  of  evil,  which 
Ave  call  natural  evil,  is  either  a necessary  consequence  of  the 


206 


CLARKE  AND  SPINOZA. 


[CHAP.  XIII. 

former,  as  death  to  a creature  on  whose  nature  immortality  was 
never  conferred ; and  then  it  is  no  more  properly  an  evil  than  the 
former.  Or  else  it  is  counterpoised  on  the  whole  with  as  great 
or  greater  good,  as  the  afflictions  and  sufferings  of  good  men, 
and  then  also  it  is  not  properly  an  evil ; or  else,  lastly,  it  is  a 
punishment,  and  then  it  is  a necessary  consequence  of  the  third 
and  last  kind  of  evil,  viz.,  moral  evil.  And  this  arises  wholly 
from  the  abuse  of  liberty  which  God  gave  to  His  creatures  for 
other  purposes,  and  which  it  was  reasonable  and  fit  to  give  them 
for  the  perfection  and  order  of  the  whole  creation.  Only  they, 
contrary  to  God’s  intention  and  command,  have  abused  what  was 
necessary  to  the  perfection  of  the  whole,  to  the  corruption  and 
depravation  of  themselves.  And  thus  all  sorts  of  evils  have  en- 
tered into  the  world  without  any  diminution  to  the  infinite  good- 
ness of  the  Creator  and  Governor  thereof.” — p.  112. 

The  main  premises  of  the  above  argument  may  be  thus 
stated : 

1st.  All  reputed  evil  is  either  evil  of  imperfection,  or  natural 
evil,  or  moral  evil. 

2nd.  Evil  of  imperfection  is  not  absolute  evil. 

3rd.  Natural  evil  is  either  a consequence  of  evil  of  imperfec- 
tion, or  it  is  compensated  with  greater  good,  or  it  is  a conse- 
quence of  moral  evil. 

4th.  That  which  is  either  a consequence  of  evil  of  imperfec- 
tion, or  is  compensated  with  greater  good,  is  not  absolute  evil. 

5th.  All  absolute  evils  are  included  in  reputed  evils. 

To  express  these  premises  let  us  assume — 

iv  = reputed  evil. 
x = evil  of  imperfection. 
y = natural  evil. 
z = moral  evil. 

p = consequence  of  evil  of  imperfection. 
q = compensated  with  greater  good. 
r = consequence  of  moral  evil. 
t = absolute  evil. 

Then,  regarding  the  premises  as  Primary  Propositions,  of  which 


CHAP.  XIII.] 


CLARKE  AND  SPINOZA. 


207 


all  the  predicates  are  particular,  and  the  conjunctions  either , or, 
as  absolutely  disjunctive,  we  have  the  following  equations  : 

w=  v {ar(l -y)  (1  - q)  +y(l  -x)  (1  -z)  + z{l  - x)  (1  - y)) 

X = V (1  - t). 

y = v{p(l-q)  (1-r)  + <7(1  - />)  (1  - r)  +r(l  -p)  (1  - q)} 

p - q)  + q(l  - p)  = v - t ). 
t = vw. 

From  which,  if  we  separately  eliminate  the  symbol  v,  we  have 
w {l  - x (1  -y)  (1  - z)  - y (l  -x)  (1  - z)  -z  (1  - x)  (l  - y )}  =0,(1) 

xt  = 0,  (2) 

y{l-p(l-q)(\-r)-q(\-p){\-r)-r(l-p)  (l-?)}=0,  (3) 

(H1  -q)  + ?(1  ~P)}  t=  °,  (4) 

^(1  - w)  = 0.  (5) 

Let  it  be  required,  first,  to  find  what  conclusion  the  premises 
warrant  us  in  forming  respecting  absolute  evils,  as  concerns  their 
dependence  upon  moral  evils,  and  the  consequences  of  moral 
evils. 

For  this  purpose  we  must  determine  t in  terms  of  z and  r. 
The  symbols  w,  x,  y,  p,  q must  therefore  be  eliminated.  The 
process  is  easy,  as  any  set  of  the  equations  is  reducible  to  a single 
equation  by  addition. 

Eliminating  w from  (1)  and  (5),  we  have 

t[l-x(\-y)(l-z)-y  (1-®)(1  -z)-z(l-x)(\-y))  = 0.  (6) 
The  elimination  of  p from  (3)  and  (4)  gives 

yqr  + yqt  + yt{i-r)(\-q)  = o.  (7) 

The  elimination  of  q from  this  gives 

yt{\  - r)  = 0.  (8) 

The  elimination  of  x between  (2)  and  (6)  gives 

*{y*+  (i-y)  0 -*)}  = °-  (9) 

The  elimination  of  y from  (8)  and  (9)  gives 
t{l-z)  (1  - r)  = 0. 

This  is  the  only  relation  existing  between  the  elements  t , z,  and  r. 


208 


CLARKE  AND  SPINOZA. 


[chap.  XIII. 


We  hence  get 

,= 0 

(1  -z)  (1  - r) 

= ^r  + ^(1-r)  + jj(I-2)?-  + 0(1-2)(1-r) 

0 0 

~03+0^  ~ z)r> 

the  interpretation  of  which  is,  Absolute  evil  is  either  moral  evil , or 
it  is,  if  not  moral  evil,  a consequence  of  moral  evil. 

Any  of  the  results  obtained  in  the  process  of  the  above  solu- 
tion furnish  us  with  interpretations.  Thus  from  (8)  we  might 
deduce 

* ■ Jrr7)  - 5 ’Jr  + °o  ( 1 -■ S') r + 5 0 1 - C 1 - r> 

o 0 ,,  . 

-o^  + o(1*»); 

Avhence,  Absolute  evils  are  either  natural  evils,  which  are  the  con- 
sequences of  moral  evils,  or  they  are  not  natural  evils  at  all. 

A variety  of  other  conclusions  may  be  deduced  from  the  given 
equations  in  reply  to  questions  which  may  be  arbitrarily  pro- 
posed. Of  such  I shall  give  a few  examples,  without  exhibiting 
the  intermediate  processes  of  solution. 

Quest.  1. — Can  any  relation  be  deduced  from  the  premises 
connecting  the  following  elements,  viz. : absolute  evils,  conse- 
quences of  evils  of  imperfection,  evils  compensated  with  greater 
good  ? 

Ans. — No  relation  exists.  If  we  eliminate  all  the  symbols  but 
z,  p,  q,  the  result  is  0 = 0. 

Quest.  2. — Is  any  relation  implied  between  absolute  evils, 
evils  of  imperfection,  and  consequences  of  evils  of  imperfection. 
Ans. — The  final  relation  between  x,  t,  and  p is 

xt  + pt  = 0 ; 

whence 

t = — — = (1  -p)  (1  -*). 

p + x 0 N /v 

Therefore,  Absolute  evils  are  neither  evils  of  imperfection,  nor  con- 
sequences of  evils  of  imperfection. 


CLARKE  AND  SPINOZA. 


209 


CHAP.  XIII.] 


Quest.  3. — Required  the  relation  of  natural  evils  to  evils  of 

imperfection  and  evils  compensated  with  greater  good. 

We  find  n 

pqij  = 0, 


0 0 \ 0 / 1 \ 

Therefore,  Natural  evils  are  either  consequences  of  evils  of  imper- 
fection which  are  not  compensated  with  greater  good , or  they  are  not 
consequences  of  evils  of  imperfection  at  all. 

Quest.  4. — In  what  relation  do  those  natural  evils  which  are 
not  moral  evils  stand  to  absolute  evils  and  the  consequences  of 
moral  evils  ? 

If  y (1  - z)  = s,  we  find,  after  elimination, 
ts  ( 1 - r)  = 0 ; 

0 0 0 ,, 

t (1  - r)  0 0 v ’ 

Therefore,  Natural  evils , ivhich  are  not  moral  evils , are  either  abso- 
lute evils , which  are  the  consequences  of  moral  evils,  or  they  are  not 
absolute  evils  at  all. 

The  following  conclusions  have  been  deduced  in  a similar 
manner.  The  subject  of  each  conclusion  will  show  of  what  par- 
ticular things  a description  was  required,  and  the  predicate  will 
show  what  elements  it  was  designed  to  involve : — 

Absolute  evils,  which  are  not  consequences  of  moral  evils,  are 
moral  and  not  natural  evils. 

Absolute  evils  which  are  not  moral  evils  are  natural  evils,  ivhich 
are  the  consequences  of  moral  evils. 

Natural  evils  which  are  not  consequences  of  moral  evils  are  not 
absolute  evils. 

Lastly,  let  us  seek  a description  of  evils  which  are  not  abso- 
lute, expressed  in  terms  of  natural  and  moral  evils. 

We  obtain  as  the  final  equation, 

1 - t = i/z  + Jy  0-  ~z)  + ^(l  -y)2  + C1  -y)  0 ~z)- 

The  direct  interpretation  of  this  equation  is  a necessary  truth, 
but  the  reverse  interpretation  is  remarkable.  Evils  ivhich  are  both 


CLARKE  AND  SPINOZA. 


210 


[chap.  XIII. 


natural  and  moral,  and  evils  which  are  neither  natural  nor  moral, 
are  not  absolute  evils. 

This  conclusion,  though  it  may  not  express  a truth,  is  cer- 
tainly involved  in  the  given  premises,  as  formally  stated. 

15.  Let  us  take  from  the  same  argument  a somewhat  fuller 
system  of  premises,  and  let  us  in  those  premises  suppose  that  the 
particles,  either , or,  are  not  absolutely  disjunctive,  so  that  in  the 
meaning  of  the  expression,  “ either  evil  of  imperfection,  or  na- 
tural evil,  or  moral  evil,”  we  include  whatever  possesses  one  or 
more  of  these  qualities. 

Let  the  premises  be  — 

1.  All  evil  ( w ) is  either  evil  of  imperfection  (x),  or  natural 
evil  (y),  or  moral  evil  (z). 

2.  Evil  of  imperfection  ( x ) is  not  absolute  evil  If). 

3.  Natural  evil  ( y ) is  either  a consequence  of  evil  of  imper- 
fection (p),  or  it  is  compensated  with  greater  good  (q),  or  it  is  a 
consequence  of  moral  evil  (r). 

4.  Whatever  is  a consequence  of  evil  of  imperfection  ( p ) is 
not  absolute  evil  ( [t ). 

5.  Whatever  is  compensated  with  greater  good  (q)  is  not 
absolute  evil  (t). 

6.  Moral  evil  (z)  is  a consequence  of  the  abuse  of  liberty  (u). 

7.  That  which  is  a consequence  of  moral  evil  (r)  is  a conse- 
quence of  the  abuse  of  liberty  ( u ). 

8.  Absolute  evils  are  included  in  reputed  evils. 

The  premises  expressed  in  the  usual  way  give,  after  the  elimi- 
nation of  the  indefinite  symbols  v,  the  following  equations  : 


© 

II 

i 

1 

1“^ 

'hs 

1 

H 

g 

(1) 

xt  = 0, 

(2) 

<< 

i 

1 

h— 

1 

II 

© 

(3) 

pt  = 0, 

(4) 

qt  = 0, 

(5) 

z (1  - u)  = 0, 

(6) 

r (1  - «)  = 0, 

(7) 

t (1  - w)  = 0. 

(8) 

Each  of  these  equations  satisfies  the  condition  F(1  - V)  = 0. 


CHAP.  XIII.] 


CLARKE  AND  SPINOZA. 


211 


The  following  results  are  easily  deduced — 

Natural  evil  is  either  absolute  evil , which  is  a consequence  of  mo- 
ral evil , or  it  is  not  absolute  evil  at  all. 

All  evils  are  either  absolute  evils,  which  are  consequences  of  the 
abuse  of  liberty,  or  they  are  not  absolute  evils. 

Natural  evils  are  either  evils  of  imperfection,  which  are  not  ab- 
solute evils,  or  they  are  not  evils  of  imperfection  at  all. 

Absolute  evils  are  either  natural  evils,  which  are  consequences  of 
the  abuse  of  liberty,  or  they  are  not  natural  evils,  and  at  the  same 
time  not  evils  of  imperfection. 

Consequences  of  the  abuse  of  liberty  include  all  natural  evils 
which  are  absolute  evils,  and  are  not  evils  of  imperfection , with  an 
indefinite  remainder  of  natural  evils  which  are  not  absolute,  and  of 
evils  which  are  not  natural. 

16.  These  examples  Avill  suffice  for  illustration.  The  reader 
can  easily  supply  others  if  they  are  needed.  W e proceed  now  to 
examine  the  most  essential  portions  of  the  demonstration  of 
Spinoza. 

DEFINITIONS. 

1.  By  a cause  of  itself  (causa  sui ),  I understand  that  of  which 
the  essence  involves  existence,  or  that  of  which  the  nature  can- 
not be  conceived  except  as  existing. 

2.  That  thing  is  said  to  be  finite  or  bounded  in  its  own  kind 
(in  suo  genere  finita)  which  may  be  bounded  by  another  thing  of 
the  same  kind ; e.  g.  Body  is  said  to  be  finite,  because  we  can 
always  conceive  of  another  body  greater  than  a given  one.  So 
thought  is  bounded  by  other  thought.  But  body  is  not  bounded 
by  thought,  nor  thought  by  body. 

3.  By  substance,  I understand  that  which  is  in  itself  (in  se), 
and  is  conceived  by  itself  ( per  se  concipitur),  i.  e.,  that  whose 
conception  does  not  require  to  be  formed  from  the  conception  of 
another  thing. 

4.  By  attribute,  I understand  that  which  the  intellect  per- 
ceives in  substance,  as  constituting  its  very  essence. 

5.  By  mode,  I understand  the  affections  of  substance,  or  that 
which  is  in  another  thing,  by  which  thing  also  it  is  conceived. 

6.  By  God,  I understand  the  Being  absolutely  infinite,  that 


CLARKE  AND  SPINOZA. 


212 


[chap.  XIII. 


is  the  substance  consisting  of  infinite  attributes,  each  of  which 
expresses  an  eternal  and  infinite  essence. 

Explanation. — I say  absolutely  infinite,  not  infinite  in  its 
own  kind.  For  to  whatever  is  only  infinite  in  its  own  kind  we 
may  deny  the  possession  of  (some)  infinite  attributes.  But  when 
a thing  is  absolutely  infinite,  whatsoever  expresses  essence  and 
involves  no  negation  belongs  to  its  essence. 

7.  That  thing  is  termed  free , which  exists  by  the  sole  neces- 
sity of  its  own  nature,  and  is  determined  to  action  by  itself  alone ; 
necessary , or  rather  constrained,  which  is  determined  by  another 
thing  to  existence  and  action,  in  a certain  and  determinate  man- 
ner. 

8.  By  eternity,  I understand  existence  itself,  in  so  far  as  it  is 
conceived  necessarily  to  follow  from  the  sole  definition  of  the 
eternal  thing. 

Explanation. — For  such  existence,  as  an  eternal  truth,  is  con- 
ceived as  the  essence  of  the  thing,  and  therefore  cannot  be  ex- 
plained by  mere  duration  or  time,  though  the  latter  should  be 
conceived  as  without  beginning  and  without  end. 


AXIOMS. 

1 . All  things  which  exist  are  either  in  themselves  (in  se)  or 
in  another  thing. 

2.  That  which  cannot  be  conceived  by  another  thing  ought 
to  be  conceived  by  itself. 

3.  From  a given  determinate  cause  the  effect  necessarily  fol- 
lows, and,  contrariwise,  if  no  determinate  cause  be  granted,  it  is 
impossible  that  an  effect  should  follow. 

4.  The  knowledge  of  the  effect  depends  upon,  and  involves, 
the  knowledge  of  the  cause. 

5.  Things  which  have  nothing  in  common  cannot  be  under- 
stood by  means  of  each  other ; or  the  conception  of  the  one  does 
not  involve  the  conception  of  the  other. 

6.  A true  idea  ought  to  agree  with  its  own  object.  ( Idea 
vera  debet  cum  suo  ideato  convenire.) 

7.  Whatever  can  be  conceived  as  non-existing  does  not  in- 
volve existence  in  its  essence. 


CLARKE  AND  SPINOZA. 


213 


CHAP.  XIII.] 

Other  definitions  are  implied,  and  other  axioms  are  virtually 
assumed,  in  some  of  the  demonstrations.  Thus,  in  Prop.  I., 
“ Substance  is  prior  in  nature  to  its  affections,”  the  proof  of 
which  consists  in  a mere  reference  to  Defs.  3 and  5,  there  seems 
to  be  an  assumption  of  the  following  axiom,  viz.,  “ That  by  which 
a thing  is  conceived  is  prior  in  nature  to  the  thing  conceived.” 
Again,  in  the  demonstration  of  Prop.  v.  the  converse  of  this 
axiom  is  assumed  to  be  true.  Many  other  examples  of  the  same 
kind  occur.  It  is  impossible,  therefore,  by  the  mere  processes  of 
Logic,  to  deduce  the  whole  of  the  conclusions  of  the  first  book  of 
the  Ethics  from  the  axioms  and  definitions  which  are  prefixed  to 
it,  and  which  are  given  above.  In  the  brief  analysis  which  will 
follow,  I shall  endeavour  to  present  in- their  proper  order  what 
appear  to  me  to  be  the  real  premises,  whether  formally  stated  or 
implied,  and  shall  show  in  what  manner  they  involve  the  conclu- 
sions to  which  Spinoza  was  led. 

17.  I conceive,  then,  that  in  the  course  of  his  demonstration, 
Spinoza  effects  several  parallel  divisions  of  the  universe  of  pos- 
sible existence,  as, 

1st.  Into  things  which  are  in  themselves,  x,  and  things  which 
are  in  some  other  tiling,  x ; whence,  as  these  classes  of  thing  toge- 
ther make  up  the  universe,  we  have 

x + x'  = 1 ; (Ax.  i.) 
or,  x = 1 - x'. 

2nd.  Into  things  which  are  conceived  by  themselves,  y,  and 
things  which  are  conceived  through  some  other  thing,  y\ 
whence 

y =\-y'.  (Ax.  ii.) 

3rd.  Into  substance,  z,  and  modes,  z' ; whence 
z = l - zf.  (Def.  hi.  v.) 

4th.  Into  things  free,/,  and  things  necessary,/;  whence 
/=!-/•  (Def.  vii.) 

5th.  Into  things  which  are  causes  and  self-existent,  c,  and 
things  caused  by  some  other  thing,  e\  whence 

e = 1 - e.  (Def.  i.  Ax.  vii.) 


214  CLARKE  AND  SPINOZA.  [CHAP,  XIII. 

And  his  reasoning  proceeds  upon  the  expressed  or  assumed 
principle,  that  these  divisions  are  not  only  parallel,  but  equiva- 
lent. Thus  in  Def.  hi.,  Substance  is  made  equivalent  with  that 
which  is  conceived  by  itself ; whence 

z=y- 

Again,  Ax.  iv.,  as  it  is  actually  applied  by  Spinoza,  estab- 
lishes the  identity  of  cause  with  that  by  which  a tiling  is  con- 
ceived; whence 

V = e. 

Again,  in  Def.  vii.,  things  free  are  identified  with  things 
self-existent ; whence 

/=  e- 

Lastly,  in  Def.  v.,  mode  is  made  identical  with  that  which  is 
in  another  thing ; whence  z = xf,  and  therefore, 

z = x. 

All  these  results  may  be  collected  together  into  the  following 
series  of  equations,  viz.  : 

x - y = z =f  - e - \ - x = 1 - y = 1-  f - 1 - z = 1 - e. 

And  any  two  members  of  this  series  connected  together  by  the 
sign  of  equality  express  a conclusion,  whether  drawn  by  Spinoza 
or  not,  which  is  a legitimate  consequence  of  his  system.  Thus 
the  equation 

z = 1 - e, 

expresses  the  sixth  proposition  of  his  system,  viz.,  One  substance 
cannot  be  produced  by  another.  Similarly  the  equation 

z = e, 

expresses  his  seventh  proposition,  viz.,  “ It  pertains  to  the  nature 
of  substance  to  exist.”  This  train  of  deduction  it  is  unnecessary 
to  pursue.  Spinoza  applies  it  chiefly  to  the  deduction  according 
to  his  views  of  the  properties  of  the  Divine  Nature,  having  first 
endeavoured  to  prove  that  the  only  substance  is  God.  In  the 
steps  of  this  process,  there  appear  to  me  to  exist  some  fallacies, 
dependent  chiefly  upon  the  ambiguous  use  of  words,  to  which  it 
will  be  necessary  here  to  direct  attention. 


CHAP.  XIII.]  CLARKE  AND  SPINOZA.  215 

18.  In  Prop.  v.  it  is  endeavoured  to  show,  that  “ There  cannot 
exist  two  or  more  substances  of  the  same  nature  or  attribute.” 
The  proof  is  virtually  as  follows  : If  there  are  more  substances 
than  one,  they  are  distinguished  either  by  attributes  or  modes  ; 
if  by  attributes,  then  there  is  only  one  substance  of  the  same  at- 
tribute ; if  by  modes,  then,  laying  aside  these  as  non-essential, 
there  remains  no  real  ground  of  distinction.  Hence  there  exists 
but  one  substance  of  the  same  attribute.  The  assumptions  here 
involved  are  inconsistent  with  those  which  are  found  in  other 
parts  of  the  treatise.  Thus  substance,  Def.  iv.,  is  apprehended 
by  the  intellect  through  the  means  of  attribute.  By  Def.  vi.  it 
may  have  many  attributes.  One  substance  may,  therefore,  con- 
ceivably be  distinguished  from  another  by  a difference  in  some  of 
its  attributes,  while  others  remain  the  same. 

In  Prop.  viii.  it  is  attempted  to  show  that,  All  substance 
is  necessarily  infinite.  The  proof  is  as  follows.  There  ex- 
ists but  one  substance,  of  one  attribute,  Prop.  v. ; and  it  per- 
tains to  its  nature  to  exist,  Prop.  vii.  It  will,  therefore,  be  of  its 
nature  to  exist  either  as  finite  or  infinite.  But  not  as  finite,  for, 
by  Def.  n.  it  would  require  to  be  bounded  by  another  substance 
of  the  same  nature,  which  also  ought  to  exist  necessarily , Prop. 
vii.  Therefore,  there  would  be  two  substances  of  the  same 
attribute,  which  is  absurd,  Prop.  v.  Substance,  therefore,  is 
infinite. 

In  this  demonstration  the  word  “ finite”  is  confounded  with 
the  expression,  “ Finite  in  its  own  kind,”  Def.  ii.  It  is  thus  as- 
sumed that  nothing  can  be  finite,  unless  it  is  bounded  by  another 
thing  of  the  same  kind.  This  is  not  consistent  with  the  ordi- 
nary meaning  of  the  term.  Spinoza’s  use  of  the  term  finite 
tends  to  make  space  the  only  form  of  substance,  and  all  existing 
things  but  affections  of  space,  and  this,  I think,  is  really  one  of 
the  ultimate  foundations  of  his  system. 

The  first  scholium  applied  to  the  above  Proposition  is  re- 
markable. I give  it  in  the  original  words  * “ Quum  finitum  esse 
revera  sit  ex  parte  negatio,  et  infinitum  absoluta  affirmatio  exis- 
tentige  alicujus  naturas,  sequitur  ergo  ex  sola  Prop.  vii.  omnem 
substantiam  debere  esse  infinitam .”  Now  this  is  in  reality  an 
assertion  of  the  principle  affirmed  by  Clarke,  and  controverted  by 


216  CLARKE  AND  SPINOZA.  [CHAP.  XIII. 

Butler  (XIII.  11),  that  necessary  existence  implies  existence 
in  every  part  of  space.  Probably  this  principle  will  be  found  to 
lie  at  the  basis  of  every  attempt  to  demonstrate,  a priori , the 
existence  of  an  Infinite  Being. 

From  the  general  properties  of  substance  above  stated,  and 
the  definition  of  God  as  the  substance  consisting  of  infinite  at- 
tributes, the  peculiar  doctrines  of  Spinoza  relating  to  the  Divine 
Nature  necessarily  follow.  As  substance  is  self-existent,  free, 
causal  in  its  very  nature,  the  thing  in  which  other  things  are, 
and  by  which  they  are  conceived ; the  same  properties  are  also 
asserted  of  the  Deity.  He  is  self-existent,  Prop.  xi. ; indivi- 
sible, Prop.  xiii.  ; the  only  substance,  Prop.  xiv. ; the  Being  in 
which  all  things  are,  and  by  which  all  things  are  conceived, 
Prop,  xv.;  free,  Prop,  xvii.;  the  immanent  cause  of  all  things, 
Prop.  xvm.  The  proof  that  God  is  the  only  substance  is  drawn 
from  Def.  vi.,  which  is  interpreted  into  a declaration  that  “ God 
is  the  Being  absolutely  infinite,  of  whom  no  attribute  wlfich  ex- 
presses the  essence  of  substance  can  be  denied.”  Every  con- 
ceivable attribute  being  thus  assigned  by  definition  to  Him,  and 
it  being  determined  in  Prop.  v.  that  there  cannot  exist  two  sub- 
stances of  the  same  attribute,  it  follows  that  God  is  the  only 
substance. 

Though  the  “ Ethics”  of  Spinoza,  like  a large  portion  of  his 
other  writings,  is  presented  in  the  geometrical  form,  it  does  not 
afford  a good  praxis  for  the  symbolical  method  of  this  work. 
Of  course  every  train  of  reasoning  admits,  when  its  ultimate 
premises  are  truly  determined,  of  being  treated  by  that  method  ; 
but  in  the  present  instance,  such  treatment  scarcely  differs,  ex- 
cept in  the  use  of  letters  for  words,  from  the  processes  employed 
in  the  original  demonstrations.  Reasoning  which  consists  so 
largely  of  a play  upon  terms  defined  as  equivalent,  is  not  often 
met  with ; and  it  is  rather  on  account  of  the  interest  attaching  to 
the  subject,  than  of  the  merits  of  the  demonstrations,  highly  as 
by  some  they  are  esteemed,  that  I have  devoted  a few  pages 
here  to  their  exposition. 

19.  It  is  not  possible,  I think,  to  rise  from  the  perusal  of  the 
arguments  of  Clarke  and  Spinoza  without  a deep  conviction  of  the 
futility  of  all  endeavours  to  establish,  entirely  a priori,  the  existence 


CHAP.  XIII.]  CLARKE  AND  SPINOZA.  217 

of  an  Infinite  Being,  His  attributes,  and  His  relation  to  the  uni- 
verse. The  fundamental  principle  of  all  such  speculations,  viz.,  that 
whatever  we  can  clearly  conceive,  must  exist,  fails  to  accomplish 
its  end,  even  when  its  truth  is  admitted.  For  how  shall  the  finite 
comprehend  the  infinite  ? Yet  must  the  possibility  of  such  con- 
ception be  granted,  and  in  something  more  than  the  sense  of 
a mere  withdrawal  of  the  limits  of  phenomenal  existence,  before 
any  solid  ground  can  be  established  for  the  knowledge,  a priori, 
of  things  infinite  and  eternal.  Spinoza’s  affirmation  of  the  re- 
ality of  such  knowledge  is  plain  and  explicit : “ Mens  humana 
adaequatum  habet  cognitionem  agternae  et  infinite  essentia  Dei” 
(Prop,  xlvii.,  Part  2nd).  Let  this  be  compared  with  Prop, 
xxxiv.,  Part  2nd : “ Omnis  idea  quae  in  nobis  est  absoluta 
sive  adequata  et  perfect®,  vera  est and  with  Axiom  vi.,  Part 
1st,  “ Idea  vera  debet  cum  suo  ideato  convenire.”  Moreover,  this 
species  of  knowledge  is  made  the  essential  constituent  of  all  other 
knowledge : “ De  natura  rationis  est  res  sub  quadam  etemitatis 
specie  percipere”  (Prop,  xliv.,  Cor.  11.,  Part  2nd).  Were  it 
said,  that  there  is  a tendency  in  the  human  mind  to  rise  in  con- 
templation from  the  particular  towards  the  universal,  from  the 
finite  towards  the  infinite,  from  the  transient  towards  the  eternal ; 
and  that  this  tendency  suggests  to  us,  with  high  probability,  the 
existence  of  more  than  sense  perceives  or  understanding  compre- 
hends ; the  statement  might  be  accepted  as  true  for  at  least  a 
a large  number  of  minds.  There  is,  however,  a class  of  specu- 
lations, the  character  of  which  must  be  explained  in  part  by 
reference  to  other  causes, — impatience  of  probable  or  limited 
knowledge,  so  often  all  that  we  can  really  attain  to ; a desire  for 
absolute  certainty  where  intimations  sufficient  to  mark  out  before 
us  the  path  of  duty,  but  not  to  satisfy  the  demands  of  the  specu- 
lative intellect,  have  alone  been  granted  to  us  ; perhaps,  too, 
dissatisfaction  with  the  present  scene  of  things.  With  the 
undue  predominance  of  these  motives,  the  more  sober  procedure 
of  analogy  and  probable  induction  falls  into  neglect.  Yet  the  lat- 
ter is,  beyond  all  question,  the  course  most  adapted  to  our  pre- 
sent condition.  To  infer  the  existence  of  an  intelligent  cause 
from  the  teeming  evidences  of  surrounding  design,  to  rise  to  the 
conception  of  a moral  Governor  of  the  world,  from  the  study  of 


218  CLARKE  AND  SPINOZA.  [CHAP.  XIII. 

the  constitution  and  the  moral  provisions  of  our  own  nature  ; — 
these,  though  but  the  feeble  steps  of  an  understanding  limited 
in  its  faculties  and  its  materials  of  knowledge,  are  of  more  avail 
than  the  ambitious  attempt  to  arrive  at  a certainty  unattainable 
on  the  ground  of  natural  religion.  And  as  these  were  the  most 
ancient,  so  are  they  still  the  most  solid  foundations,  Revelation 
being  set  apart,  of  the  belief  that  the  course  of  this  world  is  not 
abandoned  to  chance  and  inexorable  fate. 


CHAP.  XIV.] 


EXAMPLE  OF  ANALYSIS. 


219 


CHAPTER  XIV. 

EXAMPLE  OF  THE  ANALYSIS  OF  A SYSTEM  OF  EQUATIONS  BY  THE 

METHOD  OF  REDUCTION  TO  A SINGLE  EQUIVALENT  EQUATION 

V = 0,  WHEREIN  V SATISFIES  THE  CONDITION  V (1  - V)  = 0. 

1 • X ET  us  take  the  remarkable  system  of  premises  employed 
in  the  previous  Chapter,  to  prove  that  “ Matter  is  not  a 
necessary  being and  suppressing  the  6th  premiss,  viz.,  Motion 
exists, — examine  some  of  the  consequences  which  flow  from  the 
remaining  premises.  This  is  in  reality  to  accept  as  true  Dr. 
Clarke’s  hypothetical  principles ; but  to  suppose  ourselves  igno- 
norant  of  the  fact  of  the  existence  of  motion.  Instances  may 
occur  in  which  such  a selection  of  a portion  of  the  premises  of 
an  argument  may  lead  to  interesting  consequences,  though  it  is 
with  other  views  that  the  present  example  has  been  resumed.  The 
premises  actually  employed  will  be — 

1.  If  matter  is  a necessary  being,  either  the  property  of  gravi- 
tation is  necessarily  present,  or  it  is  necessarily  absent. 

2.  If  gravitation  is  necessarily  absent,  and  the  world  is  not 
subject  to  any  presiding  intelligence,  motion  does  not  exist. 

3.  If  gravitation  is  necessarily  present,  a vacuum  is  necessary. 

4.  If  a vacuum  is  necessary,  matter  is  not  a necessary  being. 

5.  If  matter  is  a necessary  being,  the  world  is  not  subject 
to  a presiding  intelligence. 

If,  as  before,  we  represent  the  elementary  propositions  by  the 
following  notation,  viz. : 

x = Matter  is  a necessary  being. 
y = Gravitation  is  necessarily  present. 
w=  Motion  exists. 

t = Gravitation  is  necessarily  absent. 
z = The  world  is  merely  material,  and  not  subject  to  a 
presiding  intelligence. 
v •=  A vacuum  is  necessary. 


220 


EXAMPLE  OF  ANALYSIS. 


[CHAP.  XIV. 

W e shall  on  expression  of  the  premises  and  elimination  of  the 
indefinite  class  symbols  ( q ),  obtain  the  following  system  of  equa- 
tions : 

xyt  + xyt  = 0, 
tzw  = 0, 
yv  = 0, 
vx  = 0, 
xz  = 0 ; 

in  which  for  brevity  y stands  for  1 - y,  t for  1 - t,  and  so  on;  whence, 
also,  1 - t = t,  1 - y - y,  &c. 

As  the  first  members  of  these  equations  involve  only  positive 
terms,  we  can  form  a single  equation  by  adding  them  together 
(VIII.  Prop.  2),  viz. : 

xyt  + xyt  + yv  + vx  + xz  + tzio  = 0, 

end  it  remains  to  reduce  the  first  member  so  as  to  cause  it  to 
satisfy  the  condition  V ( \ - V)  = 0. 

For  this  purpose  we  will  first  obtain  its  development  with 
reference  to  the  symbols  x and  y.  The  result  is — 

(t  + v + v + z + tzw)  xy  + (t  + v + z + tzw)  xy 
+ (v  + tzw)  xy  + tzwxy  = 0. 

And  our  object  will  be  accomplished  by  reducing  the  four  coeffi- 
cients of  the  development  to  equivalent  forms,  themselves  satis- 
fying the  condition  required. 

Now  the  first  coefficient  is,  since  v + v = 1, 

1 + t + z + tzw, 

which  reduces  to  unity  (IX.  Prop.  1). 

The  second  coefficient  is 

t + v 4-  z + tzw ; 
and  its  reduced  form  (X.  3)  is 

t + tv  + tvz  + tvzw. 

The  third  coefficient,  v + tzw , reduces  by  the  same  method 
to  v +'  tzwv ; and  the  last  coefficient  tzw  needs  no  reduction. 
Hence  the  development  becomes 


CHAP.  XIV.] 


EXAMPLE  OF  ANALYSIS. 


221 


xy  + (t  + tv  + tv z + tvzw ) xy  + (y  + tzwv ) xy  + tzwxy  = 0;  (1) 

and  this  is  the  form  of  reduction  sought. 

2.  Now  according  to  the  principle  asserted  in  Prop,  hi., 
Chap,  x.,  the  whole  relation  connecting  any  particular  set  of  the 
symbols  in  the  above  equation  may  be  deduced  by  developing 
that  equation  with  reference  to  the  particular  symbols  in  question, 
and  retaining  in  the  result  only  those  constituents  whose  coef- 
ficients are  unity.  Thus,  if  ^7  and  y are  the  symbols  chosen,  we 
are  immediately  conducted  to  the  equation 

xy  = 0, 

whence  we  have 

0,  \ 

y = o (l  - x)> 

with  the  interpretation,  If  gravitation  is  necessarily  present,  mat- 
ter is  not  a necessary  being. 

Let  us  next  seek  the  relation  between  x and  w.  Developing 
(1)  with  respect  to  those  symbols,  we  get 

(y  + ty  + tvy  + tvzy  + tvzy)  xw  + (y  + ty  + tvy  + tvzy ) xw 

+ (yy  + tzvy  + tzy)  xw  + vyxw  = 0. 

The  coefficient  of  xw,  and  it  alone,  reduces  to  unity.  For 
tvzy  + tvzy  = tvy,  and  tvy  + tvy  = ty,  and  ty  + ty  = y,  and  lastly, 
y + y = 1.  This  is  always  the  mode  in  which  such  reductions 
take  place.  Hence  we  get 

xw  = 0, 

0 n x 

• • w = o 0 -*)» 

of  which  the  interpretation  is,  If  motion  exists,  matter  is  not  a ne- 
cessary being. 

If,  in  like  manner,  we  develop  (1)  with  respect  to  x and  z, 
we  get  the  equation 

xz  = 0, 

0 

* “ o 

with  the  interpretation,  If  matter  is  a necessary  being,  the  world 
is  merely  material,  and  without  a presiding  intelligence. 


222 


EXAMPLE  OF  ANALYSIS. 


[CHAP.  XIV. 


This,  indeed,  is  only  the  fifth  premiss  reproduced,  but  it 
shows  that  there  is  no  other  relation  connecting  the  two  elements 
which  it  involves. 

If  we  seek  the  whole  relation  connecting  the  elements  x,  w, 
and  y,  we  find,  on  developing  (1)  with  reference  to  those  sym- 
bols, and  proceeding  as  before, 

xy  + xwy  = 0. 


Suppose  it  required  to  determine  hence  the  consequences  of  the 
hypothesis,  “ Motion  does  not  exist,”  relatively  to  the  questions 
of  the  necessity  of  matter,  and  the  necessary  presence  of  gravita- 
tion. We  find 

- xy 
iv  = — 
xy 

. x l 0 _ 

.-.  1 - w = — = - xy  + xy  + - x ; 
xy  0 9 y 0 

or,  1 - w - xy  + x,  with  xy  - 0. 


The  direct  interpretation  of  the  first  equation  is,  If  motion  does 
not  exist,  either  matter  is  a necessary  being , and  gravitation  is  not 
necessarily  present , or  matter  is  not  a necessary  being. 

The  reverse  interpretation  is,  If  matter  is  a necessary  being, 
and  gravitation  not  necessary,  motion  does  not  exist. 

In  exactly  the  same  mode,  if  we  sought  the  full  relation  be- 
tween x,  z,  and  w , we  should  find 

xzw  + xz  = 0. 


From  this  we  may  deduce 


z 


_ 0 _ 
xw  + - X, 

0 


with  xw  = 0. 


Therefore,  If  the  world  is  merely  material,  and  not  subject  to 
any  presiding  intelligence,  either  matter  is  a necessary  being,  and 
motion  does  not  exist,  or  matter  is  not  a necessary  being. 

Also,  reversely,  If  matter  is  a necessary  being,  and  there  is  no 
such  thing  as  motion,  the  ivorld  is  merely  material. 


3.  We  might,  of  course,  extend  the  same  method  to  the  de- 


EXAMPLE  OF  ANALYSIS. 


223 


CHAP.  XIV.] 

termination  of  the  consequences  of  any  complex  hypothesis  u, 
such  as,  “ The  world  is  merely  material,  and  without  any  pre- 
siding intelligence  (z),  but  motion  exists”  (w),  with  reference  to 
any  other  elements  of  doubt  or  speculation  involved  in  the  origi- 
nal premises,  such  as,  “ Matter  is  a necessary  being”  ( x ),  “ Gra- 
vitation is  a necessary  quality  of  matter,”  ( y ).  We  should,  for 
this  purpose,  connect  with  the  general  equation  (1)  a new 
equation, 

u = wz, 

reduce  the  system  thus  formed  to  a single  equation,  V=  0,  in 
which  V satisfies  the  condition  F(1  - V)  = 0,  and  proceed  as 
above  to  determine  the  relation  between  u,  x,  and  y , and  finally  u 
as  a developed  function  of  x and  y.  But  it  is  very  much  better 
to  adopt  the  methods  of  Chapters  vm.  and  ix.  I shall  here 
simply  indicate  a few  results,  with  the  leading  steps  of  their  de- 
duction, and  leave  their  verification  to  the  reader’s  choice. 

In  the  problem  last  mentioned  we  find,  as  the  relation  con- 
necting x , y,  w,  and  z , 

xw  + xwy  + xwyz  = 0. 

And  if  we  write  u = xy,  and  then  eliminate  the  symbols  x and  y 
by  the  general  problem,  Chap,  ix.,  we  find 

xu  + xyu  = 0, 

whence  1 „ _ 0 _ 

u^-xy+Qxy  + ^x-, 

wherefore  0 _ ... 

wz  - - x with  xy  = 0. 

0 y 

Hence,  If  the  world  is  merely  material , and  without  a presiding 
intelligence , and  at  the  same  time  motion  exists,  matter  is  not  a ne- 
cessary being. 

Now  it  has  before  been  shown  that  if  motion  exists,  matter  is 
not  a necessary  being , so  that  the  above  conclusion  tells  us  even 
less  than  we  had  before  ascertained  to  be  (inferentially)  true. 
Nevertheless,  that  conclusion  is  the  proper  and  complete  answer 
to  the  question  which  was  proposed,  which  was,  to  determine 
simply  the  consequences  of  a certain  complex  hypothesis. 


224 


EXAMPLE  OF  ANALYSIS. 


[CHAP.  XIV. 

4.  It  would  thus  be  easy,  even  from  the  limited  system  of 
premises  before  us,  to  deduce  a great  variety  of  additional  infe- 
rences, involving,  in  the  conditions  which  are  given,  any  pro- 
posed combinations  of  the  elementary  propositions.  If  the  con- 
dition is  one  which  is  inconsistent  with  the  premises,  the  fact 
will  be  indicated  by  the  form  of  the  solution.  The  value  which 
the  method  will  assign  to  the  combination  of  symbols  expressive 
of  the  proposed  condition  will  be  0.  If,  on  the  other  hand,  the 
fulfilment  of  the  condition  in  question  imposes  no  restriction  upon 
the  propositions  among  which  relation  is  sought,  so  that  every 
combination  of  those  propositions  is  equally  possible, — the  fact 
will  also  be  n. Seated  by  the  form  of  the  solution.  Examples 
of  each  of  these  cas^s  ai-e  subjoined. 

If  in  the  ordinary  way  we  seek  the  consequences  which  would 
flow  from  the  condition  that  matter  is  a necessary  being,  and  at 
the  same  time  that  motion  exisis,  as  affecting  the  Propositions, 
The  world  is  merely  material,  and  without  a ■presiding  intelligence, 
and,  Gravitation  is  necessarily  present,  we  shall  obtain  the  equa- 
tion 

xw  = 0, 

which  indicates  that  the  condition  proposed  is  inconsistent  with 
the  premises,  and  therefore  cannot  be  fulfilled. 

If  we  seek  the  consequences  which  would  flow  from  the  con- 
dition that  Matter  is  not  a necessary  being,  and  at  the  same  time 
that  Motion  does  exist,  with  reference  to  the  same  elements  as 
above,  viz.,  the  absence  of  a presiding  intelligence,  and  the  neces- 
sity of  gravitation, — we  obtain  the  following  result, 

(1  - X)  W = ^ (1  -y)z  + (1  -y)  (1  - z), 

which  might  literally  be  interpreted  as  follows  : 

If  matter  is  not  a necessary  being,  and  motion  exists,  then 
either  the  world  is  merely  material  and  without  a presiding  intel- 
ligence, and  gravitation  is  necessary,  or  one  of  these  two  results  fol- 
lows without  the  other,  or  they  both  fail  of  being  true.  Wherefore 
of  the  four  possible  combinations,  of  which  some  one  is  true  of 
necessity,  and  of  which  of  necessity  one  only  can  be  true,  it  is 


EXAMPLE  OF  ANALYSIS. 


225 


CHAP.  XIV.] 

affirmed  that  any  one  may  be  true.  Such  a result  is  a truism — - 
a mere  necessary  truth.  Still  it  contains  the  only  answer  which 
can  be  given  to  the  question  proposed. 

I do  not  deem  it  necessary  to  vindicate  against  the  charge  of 
laborious  trifling  these  applications.  It  may  be  requisite  to  en- 
ter with  some  fulness  into  details  useless  in  themselves,  in  order 
to  establish  confidence  in  general  principles  and  methods.  When 
this  end  shall  have  been  accomplished  in  the  subject  of  the  pre- 
sent inquiry,  let  all  that  has  contributed  to  its  attainment,  but 
has  afterwards  been  found  superfluous,  be  forgotten. 


22C 


ARISTOTELIAN  LOGIC. 


[CHAP.  XV. 


CHAPTER  XV. 

THE  ARISTOTELIAN  LOGIC  AND  ITS  MODERN  EXTENSIONS,  EX- 
AMINED BY  THE  METHOD  OF  THIS  TREATISE. 

1 • r | ''HE  logical  system  of  Aristotle,  modified  in  its  details, 
but  unchanged  in  its  essential  features,  occupies  so  im- 
portant a place  in  academical  education,  that  some  account  of  its 
nature,  and  some  brief  discussion  of  the  leading  problems  which 
it  presents,  seem  to  be  called  for  in  the  present  work.  It  is,  I 
trust,  in  no  narrow  or  harshly  critical  spirit  that  I approach  this 
task.  My  object,  indeed,  is  not  to  institute  any  direct  compa- 
rison between  the  time-honoured  system  of  the  schools  and  that 
of  the  present  treatise  ; but,  setting  truth  above  all  other  con- 
siderations, to  endeavour  to  exhibit  the  real  nature  of  the  ancient 
doctrine,  and  to  remove  one  or  two  prevailing  misapprehensions 
respecting  its  extent  and  sufficiency. 

That  which  may  be  regarded  as  essential  in  the  spirit  and 
procedure  of  the  Aristotelian,  and  of  all  cognate  systems  of  Logic, 
is  the  attempted  classification  of  the  allowable  forms  of  inference, 
and  the  distinct  reference  of  those  forms,  collectively  or  indivi- 
dually, to  some  general  principle  of  an  axiomatic  nature,  such  as 
the  “ dictum  of  Aristotle Whatsoever  is  affirmed  or  denied  of 
the  genus  may  in  the  same  sense  be  affirmed  or  denied  of  any 
species  included  under  that  genus.  Concerning  such  general 
principles  it  may,  I thiuk,  be  observed,  that  they  either  stat 1 di- 
rectly, but  in  an  abstract  form,  the  argument  which  they  are 
supposed  to  elucidate,  and,  so  stating  that  argument,  affirm  its 
validity ; or  involve  in  their  expression  technical  terms  which, 
after  definition,  conduct  us  again  to  the  same  point,  viz., 
the  abstract  statement  of  the  supposed  allowable  forms  of  in- 
ference. The  idea  of  classification  is  thus  a pervading  element 
in  those  systems.  F urthermore,  they  exhibit  Logic  as  resolvable 
into  two  great  branches,  the  one  of  which  is  occupied  with  the 
treatment  of  categorical,  the  other  with  that  of  hypothetical  or 


ARISTOTELIAN  LOGIC. 


227 


CHAP.  XV.] 

conditional  propositions.  The  distinction  is  nearly  identical  with 
that  of  primary  and  secondary  propositions  in  the  present  work. 
The  discussion  of  the  theory  of  categorical  propositions  is,  in  all 
the  ordinary  treatises  of  Logic,  much  more  full  and  elaborate  than 
that  of  hypothetical  propositions,  and  is  occupied  partly  with 
ancient  scholastic  distinctions,  partly  with  the  canons  of  deduc- 
tive inference.  To  the  latter  application  only  is  it  necessary  to 
direct  attention  here. 

2.  Categorical  propositions  are  classed  under  the  four  fol- 
lowing heads,  viz. : 

TYPE. 

1st.  Universal  affirmative  Propositions : All  F’s  are  X’s. 

2nd.  Universal  negative  „ No  F’s  are  X’s. 

3rd.  Particular  affirmative  ,,  Some  F’s  are  Ar’s. 

4th.  Particular  negative  ,,  Some  F’s  are  not  X’s. 

To  these  forms,  four  others  have  recently  been  added,  so  as 
to  constitute  in  the  whole  eight  forms  (see  the  next  article)  sus- 
ceptible, however,  of  reduction  to  six,  and  subject  to  relations 
which  have  been  discussed  with  great  fulness  and  ability  by  Pro- 
fessor De  Morgan,  in  his  Formal  Logic.  A scheme  somewhat 
different  from  the  above  has  been  given  to  the  world  by  Sir  W. 
Hamilton,  and  is  made  the  basis  of  a method  of  syllogistic  in- 
ference, which  is  spoken  of  with  very  high  respect  by  authorities 
on  the  subject  of  Logic.* 

The  processes  of  Formal  Logic,  in  relation  to  the  above  system 
of  propositions,  are  described  as  of  two  kinds,  viz.,  “ Conversion” 
and  “ Syllogism.”  By  Conversion  is  meant  the  expression  of 
any  proposition  of  the  above  kind  in  an  equivalent  form,  but  with 
a reversed  order  of  terms.  By  Syllogism  is  meant  the  deduction 
from  two  such  propositions  having  a common  term,  whether 
subject  or  predicate,  of  some  third  proposition  inferentially  in- 
volved in  the  two,  and  forming  the  “ conclusion.”  It  is  main- 
tained by  most  writers  on  Logic,  that  these  processes,  and  ac- 
cording to  some,  the  single  process  of  Syllogism,  furnish  the 
universal  types  of  reasoning,  and  that  it  is  the  business  of  the 
mind,  in  any  train  of  demonstration,  to  conform  itself,  whether 


Thomson’s  Outlines  of  the  Laws  of  Thought,  p.  177. 


228 


ARISTOTELIAN  LOGIC. 


[CHAP.  XV. 


consciously  or  unconsciously,  to  the  particular  models  of  the  pro- 
cesses which  have  been  classified  in  the  writings  of  logicians. 

3.  The  course  which  I design  to  pursue  is  to  show  how 
these  processes  of  Syllogism  and  Conversion  may  be  conducted 
in  the  most  general  manner  upon  the  principles  of  the  present 
treatise,  and,  viewing  them  thus  in  relation  to  a system  of  Logic, 
the  foundations  of  which,  it  is  conceived,  have  been  laid  in  the 
ultimate  laws  of  thought,  to  seek  to  determine  their  true  place 
and  essential  character. 

The  expressions  of  the  eight  fundamental  types  of  proposi- 
tion in  the  language  of  symbols  are  as  follows : 


1.  All  Y’s  are  X’s, 

2.  No  Y’s  are  X’s, 

3.  Some  Y’s  are  X’s, 

4.  Some  Y’s  are  not-X’s, 

5.  All  not- Y’s  are  X’s, 

6.  No  not- Y’s  are  X’s, 

7.  Some  not-  Y’s  are  X’s, 

8.  Some  not- Y’s  are  not-X’s, 


y = vx. 
y = v(\-x). 
vy  = vx. 
vy  = v (1  - x). 

1 - y = vx.  (1) 

1 - y = v (1  - x). 
v ( 1 - y)  = vx. 
v (1  -y)  = v (1  - x). 


In  referring  to  these  forms,  it  will  be  convenient  to  apply,  in 
a sense  shortly  to  be  explained,  the  epithets  of  logical  quantity, 
“universal”  and  “particular,”  and  of  quality,  “affirmative”  and 
“ negative,”  to  the  terms  of  propositions,  and  not  to  the  propo- 
sitions themselves.  We  shall  thus  consider  the  term  “ All  Y’s,” 
as  universal-affirmative ; the  term  “ Y’s,”  or  “ Some  Y’s,”  as 
particular-affirmative  ; the  term  “ All  not-  Y’s,”  as  universal-ne- 
gative ; the  term  “ Some  not-  Y’s,”  as  particular-negative.  The 
expression  “ No  Y’s,”  is  not  properly  a term  of  a proposition,  for 
the  true  meaning  of  the  pi'oposition,  “ No  Y’s  are  X’s,”  is  “All 
Y’s  are  not-X’s.”  The  subject  of  that  proposition  is,  therefore, 
universal-affirmative,  the  predicate  particular-negative.  That 
there  is  a real  distinction  between  the  conceptions  of  “ men”  and 
“not  men”  is  manifest.  This  distinction  is  all  that  I contem- 
plate when  applying  as  above  the  designations  of  affirmative  and 
negative,  without,  however,  insisting  upon  the  etymological  pro- 
priety of  the  application  to  the  terms  of  propositions.  The 
designations  positive  and  privative  would  have  been  more  ap- 


ARISTOTELIAN  LOGIC. 


229 


CHAP.  XV.] 


propriate,  but  the  former  term  is  already  employed  in  a fixed 
sense  in  other  parts  of  this  work. 

4.  From  the  symbolical  forms  above  given  the  laws  of  con- 
version immediately  follow.  Thus  from  the  equation 

V,  = war, 

representing  the  proposition,  “ All  Y’s  are  X’s,”  we  deduce,  on 
eliminating  v, 

y (i  - x)  = o, 

which  gives  by  solution  with  reference  to  1 - x, 

1 -*  = ^(1  -y) ; 

the  interpretation  of  which  is, 

All  not-X’s  are  not-  Y’s. 

This  is  an  example  of  what  is  called  S£  negative  conversion.” 
In  like  manner,  the  equation 

y = v (1  - x). 


representing  the  proposition,  “ No  Y’s  are  X’s,”  gives 

* = ? C1 " y)» 

the  interpretation  of  which  is,  “ No  X’s  are  Y’s.”  This  is  an 
example  of  what  is  termed  simple  conversion  ; though  it  is  in  re- 
ality of  the  same  kind  as  the  conversion  exhibited  in  the  previous 
example.  All  the  examples  of  conversion  which  have  been  noticed 
by  logicians  are  either  of  the  above  kind,  or  of  that  which  con- 
sists in  the  mere  transposition  of  the  terms  of  a proposition,  with- 
out altering  their  quality,  as  when  we  change 


vy  = vx,  representing,  Some  Y’s  are  X’s, 

into 

vx  = vy,  representing,  Some  X’s  are  Y’s ; 

or  they  involve  a combination  of  those  processes  with  some  auxi- 
liary process  of  limitation,  as  when  from  the  equation 
y = vx,  representing,  All  Y’s  are  X’s, 
we  deduce  on  multiplication  by  v, 

vy  = vx,  representing,  Some  Y’s  are  X’s, 

and  hence 

vx  = vy,  representing,  Some  X’s  are  Y’s. 


230  ARISTOTELIAN  LOGIC.  [CHAP.  XV. 

In  this  example,  the  process  of  limitation  precedes  that  of 
transposition. 

From  these  instances  it  is  seen  that  conversion  is  a particu- 
lar application  of  a much  more  general  process  in  Logic,  of  which 
many  examples  have  been  given  in  this  work.  That  process  has 
for  its  object  the  determination  of  any  element  in  any  proposition, 
however  complex,  as  a logical  function  of  the  remaining  elements. 
Instead  of  confining  our  attention  to  the  subject  and  predicate, 
regarded  as  simple  terms,  we  can  take  any  element  or  any 
combination  of  elements  entering  into  either  of  them ; make  that 
element,  or  that  combination,  the  “ subject”  of  a new  proposition  ; 
and  determine  what  its  predicate  shall  be,  in  accordance  with  the 
data  afforded  to  us.  It  may  be  remarked,  that  even  the  simple 
forms  of  propositions  enumerated  above  afford  some  ground  for 
the  application  of  such  a method,  beyond  what  the  received  laws 
of  conversion  appear  to  recognise.  Thus  the  equation 

y = vx,  representing,  All  F’s  are  X’s, 


gives  us,  in  addition  to  the  proposition  before  deduced,  the  three 
following : 

1st.  y (1  - x)  = 0.  There  are  no  F’s  that  are  not- X’s. 

0 

2nd.  l-^=-a:  + (l-F).  Things  that  are  not-  F’s  include  all 

things  that  are  not-X’s,  and  an 
indefinite  remainder  of  things 
that  are  X’s. 

3rd.  x = y + ^(l  - y).  Things  that  are  X’s  include  all  things 

that  are  F’s,  and  an  indefinite 
remainder  of  things  that  are  not- 
F’s. 


These  conclusions,  it  is  true,  merely  place  the  given  propo- 
sition in  other  and  equivalent  forms, — but  such  and  no  more  is 
the  office  of  the  received  mode  of  “ negative  conversion.” 

Furthermore,  these  processes  of  conversion  are  not  elemen- 
tary, but  they  are  combinations  of  processes  more  simple  than 
they,  more  immediately  dependent  upon  the  ultimate  laws  and 
axioms  which  govern  the  use  of  the  symbolical  instrument  of 


ARISTOTELIAN  LOGIC. 


231 


CHAP.  XV.] 

reasoning.  This  remark  is  equally  applicable  to  the  case  of 
Syllogism,  which  we  proceed  next  to  consider. 

5.  The  nature  of  syllogism  is  best  seen  in  the  particular  in- 
stance. Suppose  that  we  have  the  propositions, 

All  Z’s  are  Y’s, 

All  Y’s  are  Z’s. 

From  these  we  may  deduce  the  conclusion, 

All  Z’s  are  Z’s. 

This  is  a syllogistic  inference.  The  terms  X and  Z are  called 
the  extremes,  and  Y is  called  the  middle  term.  The  function 
of  the  syllogism  generally  may  now  be  defined.  Given  two  pro- 
positions of  the  kind  whose  species  are  tabulated  in  (1),  and  in- 
volving one  middle  or  common  term  Y,  which  is  connected  in 
one  of  the  propositions  with  an  extreme  X, . in  the  other  with  an 
extreme  Z;  required  the  relation  connecting  the  extremes  X and 
Z.  The  term  Y may  appear  in  its  affirmative  form,  as,  All  Y’s, 
Some  y’s ; or  in  its  negative  form,  as,  All  not-  y’s,  Some  not- 
y’s ; in  either  proposition,  without  regard  to  the  particular  form 
which  it  assumes  in  the  other. 

Nothing  is  easier  than  in  particular  instances  to  resolve  the 
Syllogism  by  the  method  of  this  treatise.  Its  resolution  is,  in- 
deed, a particular  application  of  the  process  for  the  reduction  of 
systems  of  propositions.  Taking  the  examples  above  given, 
we  have, 

x = vy , 
y = vz; 

whence  by  substitution, 

x = vv'z, 

which  is  interpreted  into 

AH  Z’s  are  Z’s. 

Or,  proceeding  rigorously  in  accordance  with  the  method  deve- 
loped in  (VIII.  7),  we  deduce 

x (1  - y)  = 0,  y (1  - z)  = 0. 

Adding  these  equations,  and  eliminating  y,  we  have 
x (1  - z)  - 0 ; 


232 


ARISTOTELIAN  LOGIC. 


[chap.  XV. 


whence  x = - z,  or,  All  X’s  are  Z’s. 

And  in  the  same  way  may  any  other  case  be  treated. 

6.  Quitting,  however,  the  consideration  of  special  examples, 
let  us  examine  the  general  forms  to  which  all  syllogism  may  be 
reduced. 


Proposition  I. 

To  deduce  the  general  rules  of  Syllogism. 

By  the  general  rules  of  Syllogism,  I here  mean  the  rules  appli- 
cable to  premises  admitting  of  every  variety  both  of  quantity 
and  of  quality  in  their  subjects  and  predicates,  except  the  com- 
bination of  two  universal  terms  in  the  same  proposition.  The 
admissible  forms  of  propositions  are  therefore  those  of  which  a 
tabular  view  is  given  in  (1). 

Let  X and  Y be  the  elements  or  things  entering  into  the  first 
premiss,  Z and  Y those  involved  in  the  second.  Two  cases,  fun- 
damentally different  in  character,  will  then  present  themselves. 
The  terms  involving  Y will  either  be  of  like  or  of  unlike  quality, 
those  terms  being  regarded  as  of  like  quality  when  they  both 
speak  of  “ Y’s,”  or  both  of  “ Not-  Y’ s,”  as  of  unlike  quality  when 
one  of  them  speaks  of  “ Y’s,”  and  the  other  of  “ Not-  Y’s.”  Any 
pair  of  premises,  in  which  the  former  condition  is  satisfied,  may 
be  represented  by  the  equations 

vx  = v'y,  (1) 

wz  = wy  ; (2) 

for  we  can  employ  the  symbol  y to  represent  either  “ All  Y’s,” 
or  “ All  not-  Y’s,”  since  the  interpretation  of  the  symbol  is  purely 
conventional.  If  we  employ  y in  the  sense  of  “ All  not-  Y’s,” 
then  1 -y  will  represent  “All  Y’s,”  and  no  other  change  will 
be  introduced.  An  equal  freedom  is  permitted  with  respect 
to  the  symbols  x and  z , so  that  the  equations  (1)  and  (2)  may, 
by  properly  assigning  the  interpretations  of  x,  y,  and  z,  be  made 
to  repi'esent  all  varieties  in  the  combination  of  premises  depen- 
dent upon  the  quality  of  the  respective  terms.  Again,  by  as- 
suming proper  interpretations  to  the  symbols  v,  v1,  w,  w',  in  those 
equations,  all  varieties  with  reference  to  quantity  may  also  be 


CHAP.  XV.] 


ARISTOTELIAN  LOGIC. 


233 


represented.  Thus,  if  we  take  v=  1,  and  represent  by  v a class 
indefinite,  the  equation  (1)  will  represent  a universal  proposition 
according  to  the  ordinary  sense  of  that  term,  i.  e.,  a proposition 
with  universal  subject  and  particular  predicate.  We  may,  in 
fact,  give  to  subject  and  predicate  in  either  premiss  whatever 
quantities  (using  this  term  in  the  scholastic  sense)  we  please,  ex- 
cept that  by  hypothesis,  they  must  not  both  be  universal.  The 
system  (1),  (2),  represents,  therefore,  with  perfect  generality, 
the  possible  combinations  of  premises  which  have  like  middle 
terms. 

7-  That  our  analysis  may  be  as  general  as  the  equations  to 
which  it  is  applied,  let  us,  by  the  method  of  this  work,  elimi- 
nate y from  (1)  and  (2),  and  seek  the  expressions  for  x,  1 - x,  and 
vx,  in  terms  of  0 and  of  the  symbols  v,  v,  w,  iv'.  The  above  will 
include  all  the  possible  forms  of  the  subject  of  the  conclusion. 
The  form  v (1  -x)  is  excluded,  inasmuch  as  we  cannot  from  the 
interpretation  vx  = Some  X’s,  given  in  the  premises,  interpret 
v (1  - x)  as  Some  not-X’s.  The  symbol  v,  when  used  in  the  sense 
of  “some,”  applies  to  that  term  only  with  which  it  is  connected 
in  the  premises. 

The  results  of  the  analysis  are  as  follows  : 
x = [yv'ivw1  + {w/(l -w)  (l-t«')+2i;w'(l-'y)(l-r')+(l-r)(l -w))]z 

+ ||  {w/(l-tr')+l-®}  (1  -z),  (I.) 


1 -x  = \y  (1  - v)  {ww  + (1-m>)(1-  iv '))  + v (1  - ui)  iv 
+ ^ {ot'(1  - iv)  (1  — «/)  + ww’  (1  - u)  (1  -i/)  + (l  - r)  (1  - m>)}]2 

+ [u  (1  - w)  w + ^ \vv'  (1  - iv')  + 1 - vj]  (1  - z),  (II.) 

vx  = { vvww  + vv  (1  - w)  (1  - tt/)}  z + ^ (1  - W~)  (1  -Z).  (III.) 


Each  of  these  expressions  involves  in  its  second  member  two 
terms,  of  one  of  which  z is  a factor,  of  the  other  1-2.  But 
syllogistic  inference  does  not,  as  a matter  of  form,  admit  of  con- 
trary classes  in  its  conclusion,  as  of  Z’s  and  not-X’s  together. 


234 


ARISTOTELIAN  LOGIC. 


[CHAP.  XV. 


We  must,  therefore,  in  order  to  determine  the  rules  of  that 
species  of  inference,  ascertain  under  what  conditions  the  second 
members  of  any  of  our  equations  are  reducible  to  a single  term. 

The  simplest  form  is  (III.),  and  it  is  reducible  to  a single 
term  if  w = 1 . The  equation  then  becomes 

vx  = vv'wz , (3) 

the  first  member  is  identical  with  the  extreme  in  the  first  pre- 
miss; the  second  is  of  the  same  quantity  and  quality  as  the  extreme 
in  the  second  premiss.  For  since  w'  = 1,  the  second  member  of 
(2),  involving  the  middle  term  y,  is  universal ; therefore,  by  the 
hypothesis,  the  first  member  is  particular,  and  therefore,  the  se- 
cond member  of  (3),  involving  the  same  symbol  w in  its  coeffi- 
cient, is  particular  also.  Hence  we  deduce  the  following  law. 

Condition  of  Inference. — One  middle  term,  at  least,  uni- 
versal. 

Rule  of  Inference. — Equate  the  extremes. 

From  an  analysis  of  the  equations  (I.)  and  (II.),  it  will  further 
appear,  that  the  above  is  the  only  condition  of  syllogistic  in- 
ference when  the  middle  terms  are  of  like  quality.  Thus  the 
second  member  of  (I.)  reduces  to  a single  term,  if  w = 1 and 
v = 1 ; and  the  second  member  of  (II.)  reduces  to  a single  term, 
if  w = 1,  v = 1,  w = 1 . In  each  of  these  cases,  it  is  necessary  that 
w'  = 1 , the  solely  sufficient  condition  before  assigned. 

Consider,  secondly,  the  case  in  which  the  middle  terms  are 
of  unlike  quality.  The  premises  may  then  be  represented  un- 
der the  forms 

vx  = v'y,  (4) 

wz  - w ( 1 - y)  ; (5) 

and  if,  as  before,  we  eliminate  y,  and  determine  the  expressions 
of  x,  1 - x,  and  vx,  we  get 

x = [wu'(l  - w)  w'  + H [ww'(  1 -v)  + (1  -v)  (1  - v)  (1  - w) 

+ v'  (1  - w)  (1  - m/))]s 

+ [twV+2  {(l-o)  (1-0  + F(1-  «0)](l-z).  (IV.) 


CHAP.  XV.]  ARISTOTELIAN  LOGIC.  235 

1 - x = \wiov  + V (1  - v')  (1  - w)  + ^ [ww  (1  - V ) 

+ (1  - v)  (1  - v)  (1  - w)  + v'(l  - w ) (1  - «/) j]  z 

+ [»(1  - *0  + II  w - 0 -«)  (x -”')}]  C1"2)-  (V.) 

vx  = {uu'(l  -w)w  + jj  vv  (\  - to)  (1  - w )}  z 

+ [vv'w  + ^ W(1  - ti/)}  (1  - z ).  (VI.) 

Now  the  second  member  of  (VI.)  reduces  to  a single  term  rela- 
tively to  zy  if  w = 1,  giving 

TO  = { w'w'  + ^ VV  ■(1-w/)}  (1-2); 

the  second  member  of  which  is  opposite,  both  in  quantity  and 
quality,  to  the  corresponding  extreme,  ivz,  in  the  second  premiss. 
For  since  w = 1,  wz  is  universal.  But  the  factor  vv'  indicates 
that  the  term  to  which  it  is  attached  is  particular,  since  by  hypo- 
thesis v and  v are  not  both  equal  to  1 . Hence  we  deduce  the 
following  law  of  inference  in  the  case  of  like  middle  terms  : 

First  Condition  of  Inference. — At  least  one  universal 
extreme. 

Rule  of  Inference. — Change  the  quantity  and  quality  of 
that  extreme,  and  equate  the  result  to  the  other  extreme. 

Moreover,  the  second  member  of  (V.)  reduces  to  a single  term 
if  v'  *=  1,  w'  = 1 ; it  then  gives 

1 - x = [vw  + ^ (1  - v)  to}  z. 

Now  since  v =1,  w' =\,  the  middle  terms  of  the  premises  are 
both  universal,  therefore  the  extremes  vx,  wz,  are  particular. 
But  in  the  conclusion  the  extreme  involving  x is  opposite,  both 
in  quantity  and  quality,  to  the  extreme  vx  in  the  first  premiss, 
while  the  extreme  involving  2 agrees  both  in  quantity  and  qua- 
lity with  the  corresponding  extreme  wz  in  the  second  premiss. 
Hence  the  following  general  law : 


236  ARISTOTELIAN  LOGIC.  [CHAP.  XV. 

Second  Condition  of  Inference. — Two  universal  middle 
terms. 

Rule  of  Inference. — Change  the  quantity  and  quality  of 
either  extreme , and  equate  the  result  to  the  other  extreme  un- 
changed. 

There  are  in  the  case  of  unlike  middle  terms  no  other  condi- 
tions or  rules  of  syllogistic  inference  than  the  above.  Thus  the 
equation  (IV.),  though  reducible  to  the  form  of  a syllogistic  con- 
clusion, when  w = 1 and  v = 1 , does  not  thereby  establish  a new 
condition  of  inference ; since,  by  what  has  preceded,  the  single 
condition  v = 1,  or  iv  = 1,  would  suffice. 

8.  The  following  examples  will  sufficiently  illustrate  the  ge- 
neral rules  of  syllogism  above  given  : 

1 . All  Y s are  X’s. 

All  Z’s  are  Y’s. 

This  belongs  to  Case  1 . All  Y’s  is  the  universal  middle  term. 
The  extremes  equated  give  as  the  conclusion 
All  Z’s  are  X’s ; 

the  universal  term,  All  Z’s,  becoming  the  subject;  the  particular 
term  (some)  X’s,  the  predicate. 

2.  All  X’s  are  Y’s. 

No  Z’s  are  Y s. 

The  proper  expression  of  these  premises  is 
All  X’s  are  Y s. 

All  Z’s  are  not-  Y’s. 

They  belong  to  Case  2,  and  satisfy  the  first  condition  of  inference. 
The  middle  term,  Y’s,  in  the  first  premiss,  is  particular-affirma- 
tive ; that  in  the  second  premiss,  not-  Y’ s,  particular-negative. 
If  we  take  All  Z’s  as  the  universal  extreme,  and  change  its 
quantity  and  quality  according  to  the  rule,  we  obtain  the  term 
Some  not-Z’s,  and  this  equated  with  the  other  extreme,  All  X’s, 
gives, 

All  X’s  are  not-Z’s,  i.  e.,  No  X’s  are  Z’s. 

If  we  commence  with  the  other  universal  extreme,  and  proceed 
similarly,  we  obtain  the  equivalent  result, 

No  Z’s  are  X’s. 


CHAP.  XV.] 


ARISTOTELIAN  LOGIC. 


' 237 


3.  All  Fs  are  X’s. 

All  not-  Y’s  are  Z’s. 

Here  also  the  middle  terms  are  unlike  in  quality.  The  premises 
therefore  belong  to  Case  2,  and  there  being  two  universal  middle 
terms,  the  second  condition  of  inference  is  satisfied.  If  by  the 
rule  we  change  the  quantity  and  quality  of  the  first  extreme, 
(some)  X’s,  we  obtain  All  not- X’s,  which,  equated  with  the 
other  extreme,  gives 

All  not- X’s  are  Z’s. 

The  reverse  order  of  procedure  would  give  the  equivalent  result, 
All  not-Z’s  are  X’s. 

The  conclusions  of  the  two  last  examples  would  not  be  recog- 
nised as  valid  in  the  scholastic  system  of  Logic,  which  virtually 
requires  that  the  subject  of  a proposition  should  be  affirmative. 
They  are,  however,  perfectly  legitimate  in  themselves,  and  the 
rules  by  which  they  are  determined  form  undoubtedly  the  most 
general  canons  of  syllogistic  inference.  The  process  of  investi- 
gation by  which  they  are  deduced  will  probably  appear  to  be  of 
needless  complexity ; and  it  is  certain  that  they  might  have  been 
obtained  with  greater  facility,  and  without  the  aid  of  any  sym- 
bolical instrument  whatever.  It  was,  however,  my  object  to 
conduct  the  investigation  in  the  most  general  manner,  and  by  an 
anal)  sis  thoroughly  exhaustive.  With  this  end  in  view,  the 
brevity  or  prolixity  of  the  method  employed  is  a matter  of  indif- 
ferer  ze.  Indeed  the  analysis  is  not  properly  that  of  the  syllogism, 
but  i f a much  more  general  combination  of  propositions ; for  we 
are  permitted  to  assign  to  the  symbols  v,  v',  w,  io',  any  class-in- 
terpretations that  we  please.  To  illustrate  this  remark,  I will 
apply  the  solution  (I.)  to  the  following  imaginary  case : 

Suppose  that  a number  of  pieces  of  cloth  striped  with  diffe- 
rent colours  were  submitted  to  inspection,  and  that  the  two  fol- 
lowing observations  were  made  upon  them  : 

1st.  That  every  piece  striped  with  white  and  green  was  also 
striped  with  black  and  yellow,  and  vice  versa. 

2nd.  That  every  piece  striped  with  red  and  orange  was  also 
striped  with  blue  and  yelloAV,  and  vice  versa. 


238 


ARISTOTELIAN  LOGIC. 


[CHAP.  XV. 


Suppose  it  then  required  to  determine  how  the  pieces  marked 
with  green  stood  affected  with  reference  to  the  colours  white, 
black,  red,  orange,  and  blue. 

Here  if  we  assume  v = white,  x = green,  v = black,  y - yellow, 
w = red,  2 = orange,  w'  = blue,  the  expression  of  our  premises  will 
be 

vx  = vy, 
wz=  w'y. 


agreeing  with  the  system  (1)  (2).  The  equation  (I.)  then  leads 
to  the  following  conclusion : 

Pieces  striped  with  green  are  either  striped  with  orange, 
white,  black,  red,  and  blue,  together,  all  pieces  possessing  which 
character  are  included  in  those  striped  with  green ; or  they  are 
striped  with  orange,  white,  and  black,  but  not  with  red  or  blue ; 
or  they  are  striped  with  orange,  red,  and  blue,  but  not  with  white 
or  black ; or  they  are  striped  with  orange,  but  not  with  white  or 
red  ; or  they  are  striped  with  white  and  black,  but  not  with  blue 
or  orange  ; or  they  are  striped  neither  with  white  nor  orange. 

Considering  the  nature  of  this  conclusion,  neither  the  sym- 
bolical expression  (I.)  by  which  it  is  conveyed,  nor  the  analysis 
by  which  that  expression  is  deduced,  can  be  considered  as  need- 
lessly complex. 

9.  The  form  in  which  the  doctrine  of  syllogism  has  been 
presented  in  this  chapter  affords  ground  for  an  important  obser- 
vation. We  have  seen  that  in  each  of  its  two  great  divisions  the 
entire  discussion  is  reducible,  so  far,  at  least,  as  concerns  the  de- 
termination of  rules  and  methods,  to  the  analysis  of  a pair  of 
equations,  viz.,  of  the  system  (1),  (2),  when  the  premises  have 
like  middle  terms,  and  of  the  system  (4),  (5),  when  the  middle 
terms  are  unlike.  Moreover,  that  analysis  has  been  actually 
conducted  by  a method  founded  upon  certain  general  laws  de- 
duced immediately  from  the  constitution  of  language,  Chap,  n., 
confirmed  by  the  study  of  the  operations  of  the  human  mind, 
Chap,  hi.,  and  proved  to  be  applicable  to  the  analysis  of  all  sys- 
tems of  equations  whatever,  by  which  propositions,  or  combina- 
tions of  propositions,  can  be  represented,  Chap.  vni.  Here,  then, 
we  have  the  means  of  definitely  resolving  the  question,  whether 
syllogism  is  indeed  the  fundamental  type  of  reasoning, — whether 


ARISTOTELIAN  LOGIC. 


239 


CHAP.  XV.] 


the  study  of  its  laws  is  co-extensive  with  the  study  of  deductive 
logic.  For  if  it  be  so,  some  indication  of  the  fact  must  be  given 
in  the  systems  of  equations  upon  the  analysis  of  which  we  have 
been  engaged.  It  cannot  be  conceived  that  syllogism  should  be 
the  one  essential  process  of  reasoning,  and  yet  the  manifestation 
of  that  process  present  nothing  indicative  of  this  high  quality  of 
pre-eminence.  No  sign,  however,  appears  that  the  discussion  of 
all  systems  of  equations  expressing  propositions  is  involved  in 
that  of  the  particular  system  examined  in  this  chapter.  And  yet 
writers  on  Logic  have  been  all  but  unanimous  in  their  assertion, 
not  merely  of  the  supremacy,  but  of  the  universal  sufficiency  of 
syllogistic  inference  in  deductive  reasoning.  The  language  of 
Archbishop  Whately,  always  clear  and  definite,  and  on  the  sub- 
ject of  Logic  entitled  to  peculiar  attention,  is  very  express  on 
this  point.  “ For  Logic,”  he  says,  “ which  is,  as  it  were,  the 
Grammar  of  Reasoning,  does  not  bring  forward  the  regular  Syl- 
logism as  a distinct  mode  of  argumentation,  designed  to  be  substi- 
tuted for  any  other  mode ; but  as  the  form  to  which  all  correct 
reasoning  may  be  ultimately  reduced.”*  And  Mr.  Mill,  in  a 
chapter  of  his  System  of  Logic,  entitled,  “ Of  Ratiocination  or 
Syllogism,”  having  enumerated  the  ordinary  forms  of  syllogism, 
observes,  “ All  valid  ratiocination,  all  reasoning  by  which  from 
general  propositions  previously  admitted,  other  propositions, 
equally  or  less  general,  are  inferred,  may  be  exhibited  in  some  of 
the  above  forms.”  And  again:  “ We  are  therefore  at  liberty, 
in  conformity  with  the  general  opinion  of  logicians,  to  consider 
the  two  elementary  forms  of  the  first  figure  as  the  universal  types 
of  all  correct  ratiocination.”  In  accordance  with  these  views  it 
has  been  contended  that  the  science  of  Logic  enjoys  an  immunity 
from  those  conditions  of  imperfection  and  of  progress  to  which 
all  other  sciences  are  subject;!  and  its  origin  from  the  travail  of 
one  mighty  mind  of  old  has,  by  a somewhat  daring  metaphor, 
been  compared  to  the  mythological  birth  of  Pallas. 

As  Syllogism  is  a species  of  elimination,  the  question  before 
us  manifestly  resolves  itself  into  the  two  following  ones: — 1st. 
Whether  all  elimination  is  reducible  to  Syllogism ; 2ndly.  Whe- 


• Elements  of  Logic,  p.  13,  ninth  edition, 
f Introduction  to  Kant’s  “Logik.” 


240  ARISTOTELIAN  LOGIC.  [CHAP.  XV. 

tlier  deductive  reasoning  can  with  propriety  be  regarded  as  con- 
sisting only  of  elimination.  I believe,  upon  careful  examination, 
the  true  answer  to  the  former  question  to  be,  that  it  is  always 
theoretically  possible  so  to  resolve  and  combine  propositions  that 
elimination  may  subsequently  be  effected  by  the  syllogistic  ca- 
nons, but  that  the  process  of  reduction  would  in  many  instances 
be  constrained  and  unnatural,  and  would  involve  operations 
which  are  not  syllogistic.  To  the  second  question  I reply,  that 
reasoning  cannot,  except  by  an  arbitrary  restriction  of  its  mean- 
ing, be  confined  to  the  process  of  elimination.  No  definition  can 
suffice  which  makes  it  less  than  the  aggregate  of  the  methods 
which  are  founded  upon  the  laws  of  thought,  as  exercised  upon 
propositions ; and  among  those  methods,  the  process  of  elimina- 
tion, eminently  important  as  it  is,  occupies  only  a place. 

Much  of  the  error,  as  I cannot  but  regard  it,  which  prevails 
respecting  the  nature  of  the  Syllogism  and  the  extent  of  its 
office,  seems  to  be  founded  in  a disposition  to  regard  all  those 
truths  in  Logic  as  primary  which  possess  the  character  of  sim- 
plicity and  intuitive  certainty,  without  inquiring  into  the  relation 
which  they  sustain  to  other  truths  in  the  Science,  or  to  general 
methods  in  the  Art,  of  Reasoning.  Aristotle’s  dictum  de  omni  et 
nullo  is  a self-evident  principle,  but  it  is  not  found  among  those 
ultimate  laws  of  the  reasoning  faculty  to  which  all  other  laws, 
however  plain  and  self-evident,  admit  of  being  traced,  and  from 
which  they  may  in  strictest  order  of  scientific  evolution  be  de- 
duced. For  though  of  every  science  the  fundamental  truths  are 
usually  the  most  simple  of  apprehension,  yet  is  not  that  sim- 
plicity the  criterion  by  which  their  title  to  be  regarded  as  funda- 
mental must  be  judged.  This  must  be  sought  for  in  the  nature 
and  extent  of  the  structure  Avhich  they  are  capable  of  supporting. 
Taking  this  view,  Leibnitz  appears  to  me  to  have  judged  cor- 
rectly when  he  assigned  to  the  “ principle  of  contradiction”  a 
fundamental  place  in  Logic;*  for  we  have  seen  the  consequences 
of  that  law  of  thought  of  which  it  is  the  axiomatic  expression 
(III.  15).  But  enough  has  been  said  upon  the  nature  of  deduc- 
tive inference  and  upon  its  constitutive  elements.  The  subject  of 


* Nouveaux  Essais  sur  l’entendement  humain.  Liv.  IV.  cap.  2.  Theodicec 
Pt.  I.  sec.  44. 


ARISTOTELIAN  LOGIC. 


241 


CHAP.  XV.] 

induction  may  probably  receive  some  attention  in  another  part  of 
this  work. 

10.  It  has  been  remarked  in  this  chapter  that  the  ordinary 
treatment  of  hypothetical,  is  much  more  defective  than  that  of 
categorical,  propositions.  What  is  commonly  termed  the  hypo- 
thetical syllogism  appears,  indeed,  to  be  no  syllogism  at  all. 
Let  the  argument — 

If  ^4.  is  J5,  C is  D, 

But  A is  B, 

Therefore  C is  D, 

be  put  in  the  form — 

If  the  proposition  X is  true,  Y is  true, 

But  X is  true, 

Therefore  Y is  true ; 

wherein  by  X is  meant  the  proposition  A is  B,  and  by  Y,  the 
proposition  C is  D.  It  is  then  seen  that  the  premises  contain 
only  two  terms  or  elements,  while  a syllogism  essentially  involves 
three.  The  following  would  be  a genuine  hypothetical  syllogism  : 

If  A7-  is  true,  Y is  true  ; 

If  Y is  true,  Z is  true  ; 

.•.  If  Ar  is  true,  Z is  true. 

After  the  discussion  of  secondary  propositions  in  a former 
part  of  this  work,  it  is  evident  that  the  forms  of  hypothetical 
syllogism  must  present,  in  every  respect,  an  exact  counterpart  to 
those  of  categorical  syllogism.  Particular  Propositions,  such  as, 
“ Sometimes  if  X is  true,  Y is  true,”  may  be  introduced,  and  the 
conditions  and  rules  of  inference  deduced  in  this  chapter  for  ca- 
tegorical syllogisms  may,  without  abatement,  be  interpreted  to 
meet  the  corresponding  cases  in  hypothetical. 

1 1.  To  what  final  conclusions  are  we  then  led  respecting  the 
nature  and  extent  of  the  scholastic  logic?  I think  to  the  following : 
that  it  is  not  a science,  but  a collection  of  scientific  truths,  too 
incomplete  to  form  a system  of  themselves,  and  not  sufficiently 
fundamental  to  serve  as  the  foundation  upon  which  a perfect 
system  may  rest.  It  does  not,  however,  follow,  that  because  the 
logic  of  the  schools  has  been  invested  with  attributes  to  which  it 


242 


ARISTOTELIAN  LOGIC. 


[CHAI\  XV. 


has  no  just  claim,  it  is  therefore  undeserving  of  regard.  Asys- 
tcm  which  has  been  associated  with  the  very  growth  of  language, 
which  has  left  its  stamp  upon  the  greatest  questions  and  the 
most  famous  demonstrations  of  philosophy,  cannot  be  altogether 
unworthy  of  attention.  Memory,  too,  and  usage,  it  must  be  ad- 
mitted, have  much  to  do  with  the  intellectual  processes ; and 
there  are  certain  of  the  canons  of  the  ancient  logic  which  have 
become  almost  inwoven  in  the  very  texture  of  thought  in  cultured 
minds.  But  whether  the  mnemonic  forms,  in  which  the  particu- 
lar rules  of  conversion  and  syllogism  have  been  exhibited,  possess 
any  real  utility, — whether  the  very  skill  which  they  are  supposed 
to  impart  might  not,  with  greater  advantage  to  the  mental 
powers,  be  acquired  by  the  unassisted  efforts  of  a mind  left  to  its 
own  resources, — are  questions  which  it  might  still  be  not  un- 
profitable to  examine.  As  concerns  the  particular  results  de- 
duced in  this  chapter,  it  is  to  be  observed,  that  they  are  solely 
designed  to  aid  the  inquiry  concerning  the  nature  of  the  ordinary 
or  scholastic  logic,  and  its  relation  to  a more  perfect  theory  of 
deductive  reasoning. 


CHAP.  XVI.] 


OF  THE  THEORY  OF  PROBABILITIES. 


243 


CHAPTER  XVI. 

ON  THE  THEORY  OF  PROBABILITIES. 

1.  "DEFORE  the  expiration  of  another  year  just  two  centuries 
will  have  rolled  away  since  Pascal  solved  the  first  known 
question  in  the  theory  of  Probabilities,  and  laid,  in  its  solution, 
the  foundations  of  a science  possessing  no  common  share  of  the 
attraction  which  belongs  to  the  more  abstract  of  mathematical 
speculations.  The  problem  which  the  Chevalier  de  Mere,  a re- 
puted gamester,  proposed  to  the  recluse  of  Port  Royal  (not  yet 
withdrawn  from  the  interests  of  science*  by  the  more  distracting 
contemplation  of  the  “ greatness  and  the  misery  of  man”),  was 
the  first  of  a long  series  of  problems,  destined  to  call  into  exis- 
tence new  methods  in  mathematical  analysis,  and  to  render  va- 
luable service  in  the  practical  concerns  of  life.  Nor  does  the  in- 
terest of  the  subject  centre  merely  in  its  mathematical  connexion, 
or  its  associations  of  utility.  The  attention  is  repaid  which  is 
devoted  to  the  theory  of  Probabilities  as  an  independent  object 
of  speculation, — to  the  fundamental  modes  in  which  it  has  been 
conceived, — to  the  great  secondary  principles  which,  as  in  the 
contemporaneous  science  of  Mechanics,  have  gradually  been  an- 
nexed to  it, — and,  lastly,  to  the  estimate  of  the  measure  of  per- 
fection which  has  been  actually  attained.  I speak  here  of  that 
perfection  which  consists  in  unity  of  conception  and  harmony  of 
processes.  Some  of  these  points  it  is  designed  very  briefly  to 
consider  in  the  present  chapter. 

2.  A distinguished  Avriterf  has  thus  stated  the  fundamental 
definitions  of  the  science : 


• See  in  particular  a letter  addressed  by  Pascal  to  Fermat,  who  had  solicited 
his  attention  to  a mathematical  problem  (Port  Royal,  par  M.  de  Sainte  Beuve) ; 
also  various  passages  in  the  collection  of  Fragments  published  by  M.  Prosper 
Faugere. 

f Poisson,  Recherches  sur  la  Probability  des  Jugemens. 


244 


OF  THE  THEORY  OF  PROBABILITIES.  [CHAP.  XVI. 

“ The  probability  of  an  event  is  the  reason  we  have  to  believe 
that  it  has  taken  place,  or  that  it  will  take  place.” 

“ The  measure  of  the  probability  of  an  event  is  the  ratio  of 
the  number  of  cases  favourable  to  that  event,  to  the  total  num- 
ber of  cases  favourable  or  contrary,  and  all  equally  possible” 
(equally  likely  to  happen). 

From  these  definitions  it  follows  that  the  word  probability,  in 
its  mathematical  acceptation,  has  reference  to  the  state  of  our 
knowledge  of  the  circumstances  under  which  an  event  may  hap- 
pen or  fail.  With  the  degree  of  information  which  we  possess 
concerning  the  circumstances  of  an  event,  the  reason  we  have  to 
think  that  it  will  occur,  or,  to  use  a single  term,  our  expectation  of 
it,  will  vary.  Probability  is  expectation  founded  upon  partial 
knowledge.  A perfect  acquaintance  with  all  the  circumstances 
affecting  the  occurrence  of  an  event  would  change  expectation 
into  certainty,  and  leave  neither  room  nor  demand  for  a theory 
of  probabilities. 

3.  Though'  our  expectation  of  an  event  grows  stronger  with 
the  increase  of  the  ratio  of  the  number  of  the  known  cases  fa- 
vourable to  its  occurrence  to  the  whole  number  of  equally  pos- 
sible cases,  favourable  or  unfavourable,  it  would  be  unphilosophical 
to  affirm  that  the  strength  of  that  expectation,  viewed  as  an 
emotion  of  the  mind,  is  capable  of  being  referred  to  any  numerical 
standard.  The  man  of  sanguine  temperament  builds  high  hopes 
where  the  timid  despair,  and  the  irresolute  are  lost  in  doubt. 
As  subjects  of  scientific  inquiry,  there  is  some  analogy  between 
opinion  and  sensation.  The  thermometer  and  the  carefully  pre- 
pared photographic  plate  indicate,  not.  the  intensity  of  the  sen- 
sations of  heat  and  light,  but  certain  physical  circumstances 
which  accompany  the  production  of  those  sensations.  So  also 
the  theory  of  probabilities  contemplates  the  numerical  measure 
of  the  circumstances  upon  which  expectation  is  founded  ; and  this 
object  embraces  the  whole  range  of  its  legitimate  applications. 
The  rules  which  we  employ  in  life-assurance,  and  in  the  other 
statistical  applications  of  the  theory  of  probabilities,  are  altogether 
independent  of  the  mental  phenomena  of  expectation.  They  are 
founded  upon  the  assumption  that  the  future  will  bear  a resem- 


245 


CHAP.  XVI.]  OF  THE  THEORY  OF  PROBABILITIES. 

blance  to  the  past ; that  under  the  same  circumstances  the  same 
event  will  tend  to  recur  with  a definite  numerical  frequency ; not 
upon  any  attempt  to  submit  to  calculation  the  strength  of  human 
hopes  and  fears. 

Now  experience  actually  testifies  that  events  of  a given  species 
do,  under  given  circumstances,  tend  to  recur  with  definite  fre- 
quency, whether  their  true  causes  be  known  to  us  or  unknown. 
Of  course  this  tendency  is,  in  general,  only  manifested  when  the 
area  of  observation  is  sufficiently  large.  The  judicial  records  of 
a great  nation,  its  registries  of  births  and  deaths,  in  relation  to 
age  and  sex,  &c.,  present  a remarkable  uniformity  from  year  to 
year.  In  a given  language,  or  family  of  languages,  the  same 
sounds,  and  successions  of  sounds,  and,  if  it  be  a written  lan- 
guage, the  same  characters  and  successions  of  characters  recur 
with  determinate  frequency.  The  key  to  the  rude  Ogham  in- 
scriptions, found  in  various  parts  of  Ireland,  and  in  which  no 
distinction  of  words  could  at  first  be  traced,  was,  by  a strict  ap- 
plication of  this  principle,  recovered.*  The  same  method,  it  is 
understood,  has  been  appliedf  to  the  deciphering  of  the  cuneiform 
records  recently  disentombed  from  the  ruins  of  Nineveh  by  the 
enterprise  of  Mr.  Layard. 

4.  Let  us  endeavour  from  the  above  statements  and  defini- 
tions to  form  a conception  of  the  legitimate  object  of  the  theory 
of  Probabilities. 

Probability,  it  has  been  said,  consists  in  the  expectation 
founded  upon  a particular  kind  of  knowledge,  viz.,  the  know- 
ledge of  the  relative  frequency  of  occurrence  of  events.  Hence 
the  probabilities  of  events,  or  of  combinations  of  events,  whether 
deduced  from  a knowledge  of  the  particular  constitution  of 
things  under  which  they  happen,  or  derived  from  the  long-con- 
tinued observation  of  a past  series  of  their  occurrences  and  fai- 
lures, constitute,  in  all  cases,  our  data.  The  probability  of  some 


* The  discovery  is  due  to  the  Rev.  Charles  Graves,  Professor  of  Mathematics 

in  the  University  of  Dublin Vide  Proceedings  of  the  Royal  Irish  Academy, 

Feb.  14,  1848.  Professor  Graves  informs  me  that  he  has  verified  the  principle 
by  constructing  sequence  tables  for  all  the  European  languages, 
t By  the  learned  Orientalist,  Dr.  Edward  llincks. 


246 


OF  THE  THEORY  OF  PROBABILITIES.  [CHAP.  XVI. 

connected  event,  or  combination  of  events,  constitutes  the  cor- 
responding qucesitum,  or  object  sought.  Now  in  the  most  gene- 
ral, yet  strict  meaning  of  the  term  “ event,”  every  combination 
of  events  constitutes  also  an  event.  The  simultaneous  occur- 
rence of  two  or  more  events,  or  the  occurrence  of  an  event  under 
given  conditions,  or  in  any  conceivable  connexion  with  other 
events,  is  still  an  event.  Using  the  term  in  this  liberty  of  appli- 
cation, the  object  of  the  theory  of  probabilities  might  be  thus 
defined.  Given  the  probabilities  of  any  events,  of  whatever 
kind,  to  find  the  probability  of  some  other  event  connected  with 
them. 

5.  Events  may  be  distinguished  as  simple  or  compound,  the 
latter  term  being  applied  to  such  events  as  consist  in  a combina- 
tion of  simple  events  (I.  13).  In  this  manner  we  might  define  it 
as  the  practical  end  of  the  theory  under  consideration  to  deter- 
mine the  probability  of  some  event,  simple  or  compound,  from 
the  given  probabilities  of  other  events,  simple  or  compound, 
with  which,  by  the  terms  of  its  definition,  it  stands  connected. 

Thus  if  it  is  known  from  the  constitution  of  a die  that  there 

is  a probability,  measured  by  the  fraction  g,  that  the  result  of 

any  particular  throw  will  be  an  ace,  and  if  it  is  required  to  deter- 
mine the  probability  that  there  shall  occur  one  ace,  and  only  one, 
in  two  successive  throws,  we  may  state  the  problem  in  the  order 
of  its  data  and  its  qucesitum,  as  follows  : 

First  Datum. — Probability  of  the  event  that  the  first  throw 
will  give  an  ace  = -. 

Second  Datum. — Probability  of  the  event  that  the  second 
throw  will  give  an  ace  = g. 

Qutesitum. — Probability  of  the  event  that  either  the  first 
throw  will  give  an  ace,  and  the  second  not  an  ace ; or  the  first 
will  not  give  an  ace,  and  the  second  will  give  one. 

Here  the  two  data  are  the  probabilities  of  simple  events  de- 
fined as  the  first  throw  giving  an  ace,  and  the  second  throw 
giving  an  ace.  The  qusesitum  is  the  probability  of  a compound 
event, — a certain  disjunctive  combination  of  the  simple  events 


CHAP.  XVI.]  OF  THE  THEORY  OF  PROBABILITIES. 


247 


involved  or  implied  in  the  data.  Probably  it  will  generally  hap- 
pen, when  the  numerical  conditions  of  a problem  are  capable  of 
being  deduced,  as  above,  from  the  constitution  of  things  under 
which  they  exist,  that  the  data  will  be  the  probabilities  of  simple 
events,  and  the  qusesitum  the  probability  of  a compound  event 
dependent  upon  the  said  simple  events.  Such  is  the  case  with  a 
class  of  problems  which  has  occupied  perhaps  an  undue  share  of 
the  attention  of  those  who  have  studied  the  theory  of  probabilities, 
viz.,  games  of  chance  and  skill,  in  the  former  of  which  some 
physical  circumstance,  as  the  constitution  of  a die,  determines 
the  probability  of  each  possible  step  of  the  game,  its  issue  being 
some  definite  combination  of  those  steps ; Avhile  in  the  latter,  the 
relative  dexterity  of  the  players,  supposed  to  be  known  a priori , 
equally  determines  the  same  element.  But  where,  as  in  statisti- 
cal problems,  the  elements  of  our  knowledge  are  drawn,  not  from 
the  study  of  the  constitution  of  things,  but  from  the  registered 
observations  of  Nature  or  of  human  society,  there  is  no  reason 
why  the  data  which  such  observations  afford  should  be  the  pro- 
babilities of  simple  events.  On  the  contrary,  the  occurrence  of 
events  or  conditions  in  marked  combinations  (indicative  of  some 
secret  connexion  of  a causal  character")  suggests  to  us  the  pro- 
priety of  making  such  concurrences,  profitable  for  future  instruc- 
tion by  a numerical  record  of  their  frequency.  Now  the  data 
which  observations  of  this  kind  afford  are  the  probabilities  of 
compound  events.  The  solution,  by  some  general  method,  of 
problems  in  which  such  data  are  involved,  is  thus  not  only  essen- 
tial to  the  perfect  development  of  the  theory  of  probabilities,  but 
also  a perhaps  necessary  condition  of  its  application  to  a large 
and  practically  important  class  of  inquiries. 

6.  Before  we  proceed  to  estimate  to  what  extent  known  me- 
thods may  be  applied  to  the  solution  of  problems  such  as  the 
above,  it  will  be  advantageous  to  notice,  that  there  is  another 
form  under  which  all  questions  in  the  theory  of  probabilities  may 
be  viewed ; and  this  form  consists  in  substituting  for  events  the 
propositions  which  assert  that  those  events  have  occurred,  or 
will  occur ; and  viewing  the  element  of  numerical  probability  as 
having  reference  to  the  truth  of  those  propositions,  not  to  the  oc- 


248 


OF  THE  THEORY  OF  PROBABILITIES.  [CHAP.  XVI. 

currence  of  the  events  concerning  which  they  make  assertion. 
Thus,  instead  of  considering  the  numerical  fraction  p as  ex- 
pressing the  probability  of  the  occurrence  of  an  event  E,  let  it 
be  viewed  as  representing  the  probability  of  the  truth  of  the 
proposition  X,  which  asserts  that  the  event  E will  occur.  Si- 
milarly, instead  of  any  probability,  q,  being  considered  as  re- 
ferring to  some  compound  event,  such  as  the  concurrence  of  the 
events  E and  F,  let  it  represent  the  probability  of  the  truth  of 
the  pi’oposition  which  asserts  that  E and  F will  jointly  occur; 
and  in  like  manner,  let  the  transformation  be  made  from  disjunc- 
tive and  hypothetical  combinations  of  events  to  disjunctive  and 
conditional  propositions.  Though  the  new  application  thus  as- 
signed to  probability  is  a necessary  concomitant  of  the  old  one, 
its  adoption  will  be  attended  with  a practical  advantage  drawn 
from  the  circumstance  that  we  have  already  discussed  the  theory 
of  propositions,  have  defined  their  principal  varieties,  and  estab- 
lished methods  for  determining,  in  every  case,  the  amount  and 
character  of  their  mutual  dependence.  Upon  this,  or  upon  some 
equivalent  basis,  any  general  theory  of  probabilities  must  rest. 
I do  not  say  that  other  considerations  may  not  in  certain  cases  of 
applied  theory  be  requisite.  The  data  may  prove  insufficient  for 
definite  solution,  and  this  defect  it  may  be  thought  necessary  to 
supply  by  hypothesis.  Or,  where  the  statement  of  large  num- 
bers is  involved,  difficulties  may  arise  after  the  solution,  from  this 
source,  for  which  special  methods  of  treatment  are  required. 
But  in  eve.y  instance,  some  form  of  the  general  problem  as  above 
stated  (Art.  4)  is  involved,  and  in  the  discussion  of  that  problem 
the  proper  and  peculiar  work  of  the  theory  consists.  I desire  it 
to  be  observed,  that  to  this  object  the  investigations  of  the  fol- 
lowing chapters  are  mainly  devoted.  It  is  not  intended  to  enter, 
except  incidentally,  upon  questions  involving  supplementary  hy- 
potheses, because  it  is  of  primary  importance,  even  with  reference 
to  such  questions  (I.  17),  that  a general  method,  founded  upon 
a solid  and  sufficient  basis  of  theory,  be  first  established. 

7.  The  following  is  a summary,  chiefly  taken  from  Laplace,  of 
the  principles  which  have  been  applied  to  the  solution  of  questions 
of  probability.  They  are  consequences  of  its  fundamental  defini- 


249 


CHAP.  XVI.]  OF  THE  THEORY  OF  PROBABILITIES. 

tions  already  stated,  and  may  be  regarded  as  indicating  the  degree 
in  which  it  has  been  found  possible  to  render  those  definitions 
available. 

Principle  1st.  If p be  the  probability  of  the  occurrence  of 
any  event,  1 - p will  be  the  probability  of  its  non-occurrence. 

2nd.  The  probability  of  the  concurrence  of  two  independent 
events  is  the  product  of  the  probabilities  of  those  events. 

3rd.  The  probability  of  the  concurrence  of  two  dependent 
events  is  equal  to  the  product  of  the  probability  of  one  of  them 
by  the  probability  that  if  that  event  occur,  the  other  will  happen 
also. 

4th.  The  probability  that  if  an  event,  E,  take  place,  an  event, 
F,  will  also  take  place,  is  equal  to  the  probability  of  the  concur- 
rence of  the  events  E and  F,  divided  by  the  probability  of  the 
occurrence  of  E. 

5th.  The  probability  of  the  occurrence  of  one  or  the  other  of 
two  events  which  cannot  concur  is  equal  to  the  sum  of  their  se- 
parate probabilities. 

6th.  If  an  observed  event  can  only  result  from  some  one  of  n 
different  causes  which  are  a priori  equally  probable,  the  proba- 
bility of  any  one  of  the  causes  is  a fraction  whose  numerator  is  the 
probability  of  the  event,  on  the  hypothesis  of  the  existence  of  that 
cause;  and  whose  denominator  is  the  sum  of  the  similar  proba- 
bilities relative  to  all  the  causes. 

7th.  The  probability  of  a future  event  is  the  sum  of  the  pro- 
ducts formed  by  multiplying  the  probability  of  each  cause  by 
the  probability  that  if  that  cause  exist,  the  said  future  event 
will  take  place. 

8.  Respecting  the  extent  and  the  relative  sufficiency  of  these 
principles,  the  following  observations  may  be  made. 

1st.  It  is  always  possible,  by  the  due  combination  of  these 
principles,  to  express  the  probability  of  a compound  event,  de- 
pendent in  any  manner  upon  independent  simple  events  whose 
distinct  probabilities  are  given.  A very  large  proportion  of  the 
problems  which  have  been  actually  solved  are  of  this  kind,  and 
the  difficulty  attending  their  solution  has  not  arisen  from  the  in- 
sufficiency of  the  indications  furnished  by  the  theory  of  proba- 
bilities, but  from  the  need  of  an  analysis  which  should  render 


250 


OF  THE  THEORY  OF  PROBABILITIES.  [CHAP.  XVI. 


those  indications  available  when  functions  of  large  numbers,  or 
series  consisting  of  many  and  complicated  terms,  are  thereby  in- 
troduced. It  may,  therefore,  be  fully  conceded,  that  all  pro- 
blems having  for  their  data  the  probabilities  of  independent 
simple  events  fall  within  the  scope  of  received  methods. 

2ndly.  Certain  of  the  principles  above  enumerated,  and  espe- 
cially the  sixth  and  seventh,  do  not  presuppose  that  all  the  data 
are  the  probabilities  of  simple  events.  In  their  peculiar  applica- 
tion to  questions  of  causation,  they  do,  however,  assume,  that  the 
causes  of  which  they  take  account  are  mutually  exclusive,  so 
that  no  combination  of  them  in  the  production  of  an  etfect  is 
possible.  If,  as  before  explained,  we  transfer  the  numerical  pro- 
babilities from  the  events  with  which  they  are  connected  to  the 
propositions  by  which  those  events  are  expressed,  the  most  ge- 
neral problem  to  which  the  aforesaid  principles  are  applicable 
may  be  stated  in  the  following  order  of  data  and  qucesita. 

DATA. 

1st.  The  probabilities  of  the  n conditional  propositions  : 

If  the  cause  A i exist,  the  event  E will  follow  ; 

>>  A-i  ,,  E ,, 


An 


E 


2nd.  The  condition  that  the  antecedents  of  those  propositions 
are  mutually  conflicting. 

REQUIREMENTS. 

The  probability  of  the  truth  of  the  proposition  which  declares 
the  occurrence  of  the  event  E;  also,  when  that  proposition  is 
known  to  be  true,  the  probabilities  of  truth  of  the  several  pro- 
positions which  affirm  the  respective  occurrences  of  the  causes 
Aj , A 2 . • An . 

Here  it  is  seen,  that  the  data  are  the  probabilities  of  a series 
of  compound  events,  expressed  by  conditional  propositions.  But 
the  system  is  obviously  a very  limited  and  particular  one.  For 
the  antecedents  of  the  propositions  are  subject  to  the  condition  of 
being  mutually  exclusive,  and  there  is  but  one  consequent,  the 
event  E,  in  the  whole  system.  It  does  not  follow,  from  our 


251 


CHAP.  XVI.]  OF  THE  THEORY  OF  PROBABILITIES. 

ability  to  discuss  such  a system  as  the  above,  that  we  are  able  to 
resolve  problems  whose  data  are  the  probabilities  of  any  system 
of  conditional  propositions;  far  less  that  we  can  resolve  problems 
whose  data  are  the  probabilities  of  any  system  of  propositions 
whatever.  And,  viewing  the  subject  in  its  material  rather 
than  its  formal  aspect,  it  is  evident,  that  the  hypothesis  of  exclu- 
sive causation  is  one  which  is  not  often  realized  in  the  actual 
world,  the  phenomena  of  which  seem  to  be,  usually,  the  products 
of  complex  causes,  the  amount  and  character  of  whose  co-opera- 
tion is  unknown.  Such  is,  without  doubt,  the  case  in  nearly  all 
departments  of  natural  or  social  inquiry  in  which  the  doctrine  of 
probabilities  holds  out  any  new  promise  of  useful  applications. 

9.  To  the  above  principles  we  may  add  another,  which  has 
been  stated  in  the  following  terms  by  the  Savilian  Professor  of 
Astronomy  in  the  University  of  Oxford.* 

“ Principle  8.  If  there  be  any  number  of  mutually  exclusive 
hypotheses,  hlt  h2,  h3,  . . of  which  the  probabilities  relative  to  a 
particular  state  of  information  are  plt  p2,  p3,  . . and  if  new  infor- 
mation be  given  which  changes  the  probabilities  of  some  of  them, 
suppose  of  hm+ 1 and  all  that  follow,  without  having  otherwise 
any  reference  to  the  rest ; then  the  probabilities  of  these  latter 
have  the  same  ratios  to  one  another,  after  the  new  information, 
that  they  had  before,  that  is, 

p\  : p\  : p\  . . . : p'm  = Pi : p» : p»  • . : Pm, 

where  the  accented  letters  denote  the  values  after  the  new  infor- 
mation has  been  acquired.” 

This  principle  is  apparently  of  a more  fundamental  character 
than  the  most  of  those  before  enumerated,  and  perhaps  it  might,  as 
has  been  suggested  by  Professor  Donkin,  be  regarded  as  axio- 
matic. It  seems  indeed  to  be  founded  in  the  very  definition  of 
the  measure  of  probability,  as  “ the  ratio  of  the  number  of  cases 
favourable  to  an  event  to  the  total  number  of  cases  favourable  or 
contrary,  and  all  equally  possible.”  For,  adopting  this  definition, 
it  is  evident  that  in  whatever  proportion  the  number  of  equally 


* On  certain  Questions  relating  to  the  Theory  of  Probabilities;  by  W.  F. 
Donkin,  M.  A.,  F.  R.  S.,  &c.  Philosophical  Magazine,  May,  1851. 


252 


OF  THE  THEORY  OF  PROBABILITIES.  [CHAP.  XVI. 

possible  cases  is  diminished,  while  the'number  of  favourable  cases 
remains  unaltered,  in  exactly  the  same  proportion  will  the  pro- 
babilities of  any  events  to  which  these  cases  have  refei*ence  be 
increased.  And  as  the  new  hypothesis,  viz.,  the  diminution  of 
the  number  of  possible  cases  without  affecting  the  number  of 
them  which  are  favourable  to  the  events  in  question,  increases 
the  probabilities  of  those  events  in  a constant  ratio,  the  relative 
measures  of  those  probabilities  remain  unaltered.  If  the  principle 
we  are  considering  be  then,  as  it  appears  to  be,  inseparably  in- 
volved in  the  very  definition  of  probability,  it  can  scarcely,  of 
itself,  conduct  us  further  than  the  attentive  study  of  the  defini- 
tion would  alone  do,  in  the  solution  of  problems.  From  these 
considerations  it  appears  to  be  doubtful  whether,  without  some 
aid  of  a different  kind  from  any  that  has  yet  offered  itself  to  our 
notice,  any  considerable  advance,  either  in  the  theory  of  proba- 
bilities as  a branch  of  speculative  knowledge,  or  in  the  practical 
solution  of  its  problems  can  be  hoped  for.  And  the  establish- 
ment, solely  upon  the  basis  of  any  such  collection  of  principles  as 
the  above,  of  a method  universally  applicable  to  the  solution  of 
problems,  without  regard  either  to  the  number  or  to  the  nature 
of  the  propositions  involved  in  the  expression  of  their  data, 
seems  to  be  impossible.  For  the  attainment  of  such  an  object 
other  elements  are  needed,  the  consideration  of  which  will  occupy 
the  next  chapter. 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES. 


253 


CHAPTER  XVII. 

DEMONSTRATION  OF  A GENERAL  METHOD  FOR  THE  SOLUTION  OF 
PROBLEMS  IN  THE  THEORY  OF  PROBABILITIES. 

1.  XT  has  been  defined  (XVI.  2),  that  “the  measure  of  the 
probability  of  an  event  is  the  ratio  of  the  number  of  cases 
favourable  to  that  event,  to  the  total  number  of  cases  favourable 
or  unfavourable,  and  all  equally  possible.”  In  the  following  in- 
vestigations the  term  probability  will  be  used  in  the  above  sense 
of  “ measure  of  probability.” 

From  the  above  definition  we  may  deduce  the  following  con- 
clusions. 

I.  When  it  is  certain  that  an  event  will  occur,  the  probability 
of  that  event,  in  the  above  mathematical  sense,  is  1.  For  the 
cases  which  are  favourable  to  the  event,  and  the  cases  which  are 
possible,  are  in  this  instance  the  same. 

Hence,  also,  ifjo  be  the  probability  that  an  event  x will  happen, 
1 - p will  be  the  probability  that  the  said  event  will  not  happen. 
To  deduce  this  result  directly  from  the  definition,  let  m be  the 
number  of  cases  favourable  to  the  event  x,  n the  number  of  cases 
possible,  then  n-m  is  the  number  of  cases  unfavourable  to  the 
event  x.  Hence,  by  definition, 

771 

— = probability  that  x will  happen. 
n — in 

= probability  that  x will  not  happen. 

But  n-m  m 


II.  The  probability  of  the  concurrence  of  any  two  events  is 
the  product  of  the  probability  of  either  of  those  events  by  the 
probability  that  if  that  event  occur,  the  other  will  occur  also. 

Let  m be  the  number  of  cases  favourable  to  the  happening 
of  the  first  event,  and  n the  number  of  equally  possible  cases  un- 
favourable to  it ; then  the  probability  of  the  first  event  is,  by  defini- 


254 


GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

tion,  — — — . Of  the  m cases  favourable  to  the  first  event,  let  l 
m + n 

cases  be  favourable  to  the  conjunction  of  the  first  and  second 
events,  then,  by  definition,  — is  the  probability  that  if  the  first 

event  happen,  the  second  also  will  happen.  Multiplying  these 
fractions  together,  we  have 

mil 
m + u m m + n 

But  the  resulting  fraction  — - — has  for  its  numerator  the  num- 

m + n 

ber  of  cases  favourable  to  the  conjunction  of  events,  and  for  its 
denominator,  the  number  m + n of  possible  cases.  Therefore, 
it  represents  the  probability  of  the  joint  occurrence  of  the  two 
events. 

Hence,  if  p be  the  probability  of  any  event  x,  and  q the  pro- 
bability that  if  x occur  y will  occur,  the  probability  of  the  con- 
junction xy  will  be  pq. 

III.  The  probability  that  if  an  event  x occur,  the  event  y will 
occur,  is  a fraction  whose  numerator  is  the  probability  of  their 
joint  occurrence,  and  denominator  the  probability  of  the  occur- 
rence of  the  event  x. 

This  is  an  immediate  consequence  of  Principle  2nd. 

IV.  The  probability  of  the  occurrence  of  some  one  of  a series 
of  exclusive  events  is  equal  to  the  sum  of  their  separate  proba- 
bilities. 

For  let  n be  the  number  of  possible  cases  ; ml  the  number  of 
those  cases  favourable  to  the  first  event ; m2  the  number  of  cases 
favourable  to  the  second,  &c.  Then  the  separate  probabilities  of 

the  events  are  — , — , &c.  Again,  as  the  events  are  exclusive, 
n n 

none  of  the  cases  favourable  to  one  of  them  is  favourable  to 
another;  and,  therefore,  the  number  of  cases  favourable  to  some 
one  of  the.  seiies  will  be  m,  + m2  . . , and  the  probability  of  some 

ffl  yyi  0 . . 

one  of  the  series  happening  will  be  — — . But  this  is  the 

sum  of  the  previous  fractions,  — , &c.  Whence  the  prin- 

ciple  is  manifest. 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  255 

2.  Definition. — Two  events  are  said  to  be  independent 
when  the  probability  of  the  happening  of  either  of  them  is 
unaffected  by  our  expectation  of  the  occurrence  or  failure  of 
the  other. 

From  this  definition,  combined  with  Principle  II.,  we  have 
the  following  conclusion  : 

V.  The  probability  of  the  concurrence  of  two  independent 
events  is  equal  to  the  product  of  the  separate  probabilities  of 
those  events. 

For  if  p be  the  probability  of  an  event  x,  q that  of  an  event  y 
regarded  as  quite  independent  of  x,  then  is  q also  the  probability 
that  if  x occur  y will  occur.  Hence,  by  Principle  II.,  pq  is  the 
probability  of  the  concurrence  of  x and  y. 

Under  the  same  circumstances,  the  probability  that  x will 
occur  and  y not  occur  will  be  p (1  - q).  For  p is  the  probability 
that  x will  occur,  and  1 - q the  probability  that  y will  not  occur. 
In  like  manner  ( 1 - p)  (1  - q)  will  be  the  probability  that  both 
the  events  fail  of  occurring. 

3.  There  exists  yet  another  principle,  different  in  kind  from 
the  above,  but  necessary  to  the  subsequent  investigations  of  this 
chapter,  before  proceeding  to  the  explicit  statement  of  which  I 
desire  to  make  one  or  two  preliminary  observations. 

I would,  in  the  first  place,  remark  that  the  distinction  be- 
tween simple  and  compound  events  is  not  one  founded  in  the 
nature  of  events  themselves,  but  upon  the  mode  or  connexion  in 
which  they  are  presented  to  the  mind.  How  many  separate  par- 
ticulars, for  instance,  are  implied  in  the  terms  “ To  be  in  health,” 
“ To  prosper,”  &c.,  each  of  which  might  still  be  regarded  as 
expressing  a “ simple  event”  ? The  prescriptive  usages  of  lan- 
guage, which  have  assigned  to  particular  combinations  of  events 
single  and  definite  appellations,  and  have  left  unnumbered  other 
combinations  to  be  expressed  by  corresponding  combinations  of 
distinct  terms  or  phrases,  is  essentially  arbitrary.  When,  then, 
we  designate  as  simple  events  those  which  are  expressed  by  a 
single  verb,  or  by  what  grammarians  term  a simple  sentence,  we 
do  not  thereby  imply  any  real  simplicity  in  the  events  them- 
selves, but  use  the  term  solely  with  reference  to  grammatical 
expression. 


256  GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

4.  Now  if  this  distinction  of  events,  as  simple  or  compound,  is 
not  founded  in  their  real  nature,  but  rests  upon  the  accidents  of 
language,  it  cannot  affect  the  question  of  their  mutual  depend- 
ence or  independence.  If  my  knowledge  of  two  simple  events  is 
confined  to  this  particular  fact,  viz.,  that  the  probability  of  the 
occurrence  of  one  of  them  is  p,  and  that  of  the  other  q ; then  I re- 
gard the  events  as  independent,  and  thereupon  affirm  that  the 
probability  of  their  joint  occurrence  is  pq.  But  the  ground  of 
this  affirmation  is  not  that  the  events  are  simple  ones,  but  that 
the  data  afford  no  information  whatever  concerning  any  connexion 
or  dependence  between  them.  When  the  probabilities  of  events 
are  given,  but  all  information  respecting  their  dependence  with- 
held, the  mind  regards  them  as  independent.  And  this  mode  of 
thought  is  equally  correct  whether  the  events,  judged  according 
to  actual  expression,  are  simple  or  compound,  i.  e.,  whether  each 
of  them  is  expressed  by  a single  verb  or  by  a combination  of 
verbs. 

5.  Let  it,  however,  be  supposed  that,  together  with  the  pro- 
babilities of  certain  events,  we  possess  some  definite  information 
respecting  their  possible  combinations.  For  example,  let  it  be 
known  that  certain  combinations  are  excluded  from  happening, 
and  therefore  that  the  remaining  combinations  alone  are  possible. 
Then  still  is  the  same  general  principle  applicable.  The  mode 
in  which  we  avail  ourselves  of  this  information  in  the  calculation 
of  the  probability  of  any  conceivable  issue  of  events  depends  not 
upon  the  nature  of  the  events  whose  probabilities  and  whose 
limits  of  possible  connexion  are  given.  It  matters  not  whether 
they  are  simple  or  compound.  It  is  indifferent  from  what  source, 
or  by  what  methods,  the  knowledge  of  their  probabilities  and  of 
their  connecting  relations  has  been  derived.  We  must  regard 
the  events  as  independent  of  any  connexion  beside  that  of  which 
we  have  information,  deeming  it  of  no  consequence  whether  such  in- 
formation has  been  explicitly  conveyed  to  us  in  the  data,  or  thence 
deduced  by  logical  inference.  And  this  leads  us  to  the  statement 
of  the  general  principle  in  question,  viz.  : 

VI.  The  events  whose  probabilities  are  given  are  to  be  re- 
garded as  independent  of  any  connexion  but  such  as  is  either 
expressed,  or  necessarily  implied,  in  the  data ; and  the  mode  in 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  257 

which  our  knowledge  of  that  connexion  is  to  be  employed  is  in- 
dependent of  the  nature  of  the  source  from  which  such  know- 
ledge has  been  derived. 

The  practical  importance  of  the  above  principle  consists 
in  the  circumstance,  that  whatever  may  be  the  nature  of  the 
events  whose  probabilities  are  given, — whatever  the  nature  of 
the  event  whose  probability  is  sought,  we  are  always  able,  by  an 
application  of  the  Calculus  of  Logic,  to  determine  the  expression 
of  the  latter  event  as  a definite  combination  of  the  former  events, 
and  definitely  to  assign  the  whole  of  the  implied  relations  con- 
necting the  former  events  with  each  other.  In  other  words,  we 
can  determine  what  that  combination  of  the  given  events  is  whose 
probability  is  required,  and  what  combinations  of  them  are  alone 
possible.  It  follows  then  from  the  above  principle,  that  we  can 
reason  upon  those  events  as  if  they  were  simple  events,  whose 
conditions  of  possible  combination  had  been  directly  given  by 
experience,  and  of  which  the  probability  of  some  definite  combi- 
nation is  sought.  The  possibility  of  a general  method  in  proba- 
bilities depends  upon  this  reduction. 

6.  As  the  investigations  upon  which  we  are  about  to  enter 
are  based  upon  the  employment  of  the  Calculus  of  Logic,  it  is 
necessary  to  explain  certain  terms  and  modes  of  expression  which 
are  derived  from  this  application. 

By  the  event  x,  I mean  that  event  of  which  the  proposition 
which  affirms  the  occurrence  is  symbolically  expressed  by  the 
equation 

x - 1. 

By  the  event  <p  (x,  y,  z,  . .),  I mean  that  event  of  which  the 
occurrence  is  expressed  by  the  equation 

0(r,  y,  z,..)  = 1. 

Such  an  event  may  be  termed  a compound  event,  in  relation  to 
the  simple  events  x,  y,  z,  which  its  conception  involves.  Thus, 
if  x represent  the  event  “ It  rains,”  y the  event  “ It  thunders,” 
the  separate  occurrences  of  those  events  being  expressed  by  the 
logical  equations 

x = 1,  y = 1, 

then  will  x (1  - y)  + y (1  - x)  represent  the  event  or  state  of 


258  GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

things  denoted  by  the  Proposition,  “ It  either  rains  or  thunders, 
but  not  both the  expression  of  that  state  of  things  being 

*(i -y)  + y(i-»)  = i. 

If  for  brevity  we  represent  the  function  <p  (x,  y,  z,  . .),  used  in 
the  above  acceptation  by  V,  it  is  evident  (VI.  13)  that  the  law 
of  duality 

F(1-F)  = 0, 

will  be  identically  satisfied. 

The  simple  events  x,  y , z will  be  said  to  be  “ conditioned” 
Avhen  they  are  not  free  to  occur  in  every  possible  combination ; 
in  other  words,  when  some  compound  event  depending  upon 
them  is  precluded  from  occurring.  Thus  the  events  denoted  by 
the  propositions,  “ It  rains,”  “ It  thunders,”  are  “ conditioned” 
if  the  event  denoted  by  the  proposition,  “ It  thunders,  but  does 
not  rain,”  is  excluded  from  happening,  so  that  the  range  of  pos- 
sible is  less  than  the  range  of  conceivable  combination.  Simple 
unconditioned  events  are  by  definition  independent. 

Any  compound  event  is  similarly  said  to  be  conditioned  if  it 
is  assumed  that  it  can  only  occur  under  a certain  condition,  that 
is,  in  combination  with  some  other  event  constituting,  by  its  pre- 
sence, that  condition. 

7.  We  shall  proceed  in  the  natural  order  of  thought,  from 
simple  and  unconditioned,  to  compound  and  conditioned  events. 

Proposition  I. 

1st.  If  p,  q,  r are  the  respective  probabilities  of  any  uncon- 
ditioned simple  events  x,  y,  z , the  probability  of  any  compound 
event  V will  be  [F],  this  function  [F]  being  formed  by  changing, 
in  the  function  V,  the  symbols  x,  y,  z into  p,  q,  r,  8fC. 

2ndly.  Under  the  same  circumstances,  the  probability  that  if 
the  event  V occur , any  other  event  V will  also  occur , will  be 

, wherein  [ VVrj  denotes  the  result  obtained  by  multiplying 

together  the  logical  functions  V and  V',  and  changing  in  the  result 
x,  y,  z,  &fc.  into  p,  q,  r,  Sfc. 

Let  us  confine  our  attention  in  the  first  place  to  the  pos- 


[FF] 

m 


259 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES. 


sible  combinations  of  the  two  simple  events,  x and  y,  of  which  the 
respective  probabilities  are  p and  q.  The  primary  combinations 
of  those  events  (V.  11),  and  their  corresponding  probabilities,  are 
as  follows : 


EVENTS. 


PROBABILITIES. 


xy,  Concurrence  of  x and  y, 

x ( 1 - y),  Occurrence  of  x without  y, 

(1  - x)y,  Occurrence  of  y without  x, 

(1  - x)  (1  - y),  Conjoint  failure  of  x and  y, 


pq. 

P0--  ?)• 

0 -p)q- 

(!  -p)  (1  - q). 


We  see  that  in  these  cases  the  probability  of  the  compound  event 
represented  by  a constituent  is  the  same  function  of  p and  q as 
the  logical  expression  of  that  event  is  of  x and  y ; and  it  is  obvious 
that  this  remark  applies,  whatever  may  be  the  number  of  the 
simple  events  whose  probabilities  are  given,  and  whose  joint  ex- 
istence or  failure  is  involved  in  the  compound  event  of  which  we 
seek  the  probability. 

Consider,  in  the  second  place,  any  disjunctive  combination  of 
the  above  constituents.  The  compound  event,  expressed  in  or- 
dinary language  as  the  occurrence  of  “ either  the  event  x without 
the  event  y,  or  the  event  y without  the  event  x,”  is  symbolically 
expressed  in  the  form  x { \ - y)  + y (\  - x),  and  its  probability, 
determined  by  Principles  iv.  and  v.,  is  p (1  - q)  + q (1  - p).  The 
latter  of  these  expressions  is  the  same  function  of  p and  q as  the 
former  is  of  x and  y.  And  it  is  obvious  that  this  is  also  a par- 
ticular illustration  of  a general  rule.  The  events  which  are  ex- 
pressed by  any  two  or  more  constituents  are  mutually  exclusive. 
The  only  possible  combination  of  them  is  a disjunctive  one,  ex- 
pressed in  ordinary  language  by  the  conjunction  or , in  the  lan- 
guage of  symbolical  logic  by  the  sign  +.  Now  the  probability  of 
the  occurrence  of  some  one  out  of  a set  of  mutually  exclusive 
events  is  the  sum  of  their  separate  probabilities,  and  is  expressed 
by  connecting  the  expressions  for  those  separate  probabilities  by 
the  sign  +.  Thus  the  law  above  exemplified  is  seen  to  be  general. 
The  probability  of  any  unconditioned  event  V will  be  found  by 
changing  in  V the  symbols  x,  y,  z, . . into  p,  q,  r,  . . 

8.  Again,  by  Principle  in.,  the  probability  that  if  the  event 
V occur,  the  event  V'  will  occur  Avith  it,  is  expressed  by  a frac- 


260 


GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 


tion  whose  numerator  is  the  probability  of  the  joint  occurrence 
of  V and  V',  and  denominator  the  probability  of  the  occurrence 
of  V. 

Now  the  expression  of  that  event,  or  state  of  things,  which  is 
constituted  by  the  joint  occurrence  of  the  events  V and  V',  will 
be  formed  by  multiplying  together  the  expressions  V and  V'  ac- 
cording to  the  rules  of  the  Calculus  of  Logic ; since  whatever 
constituents  are  found  in  both  V and  V will  appear  in  the  pro- 
duct, and  no  others.  Again,  by  what  has  just  been  shown,  the 
probability  of  the  event  represented  by  that  product  will  be  de- 
termined by  changing  therein  x,  y,  z into  p,  q,  r,  . . Hence  the 
numerator  sought  will  be  what  [FF]  by  definition  represents. 
And  the  denominator  will  be  [F],  wherefore 


Probability  that  if  F occur,  V will  occur  with  it  = 


[FF] 

[r]  ' 


9.  For  example,  if  the  probabilities  of  the  simple  events 
x , y,  z are  p,  q,  r respectively,  and  it  is  required  to  find  the  pro- 
bability that  if  either  x or  y occur,  then  either  y or  z will  occur, 
we  have  for  the  logical  expressions  of  the  antecedent  and  conse- 
quent— 


1st.  Either  x or  y occurs,  x (1  - y)  + y (1  - x). 

2nd.  Either  y or  z occurs,  y (1  - z)  + z (1  - y). 

If  now  we  multiply  these  two  expressions  together  according  to 
the  rules  of  the  Calculus  of  Logic,  we  shall  have  for  the  expres- 
sion of  the  concurrence  of  antecedent  and  consequent, 

xz{\-y)  + y (\-x)  (\  - z ). 


Changing  in  this  result  x,  y,  z into  p,  q,  r,  and  similarly  trans- 
forming the  expression  of  the  antecedent,  we  find  for  the  proba- 
bility sought  the  value 

pr(l-q)  + g(  1 -p)  (1  - r) 

P(l  -?)  + <7(1  ~P) 

The  special  function  of  the  calculus,  in  a case  like  the  above,  is 
to  supply  the  office  of  the  reason  in  determining  what  are  the 
conjunctures  involved  at  once  in  the  consequent  and  the  ante- 
cedent.  But  the  advantage  of  this  application  is  almost  entirely 


261 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES. 

prospective,  and  will  be  made  manifest  in  a subsequent  propo- 
sition. 

Proposition  II. 

10.  It  is  known  that  the  probabilities  of  certain  simple  events 
x,  y,  z,  . . are  p,  q,r,  . . respectively  when  a certain  condition  V is 
satisfied ; V being  in  expression  a function  of  x,  y,  z,  . . Required 
the  absolute  probabilities  of  the  events  x,  y,  z,  . . , that  is,  the 
probabilities  of  their  respective  occurrence  independently  of  the  con- 
dition V. 


Let p,  q',  r,  &c.,  be  the  probabilities  required,  i.  e.  the  pro- 
babilities of  the  events  x,  y,  z,  . . , regarded  not  only  as  simple, 
but  as  independent  events.  Then  by  Prop.  i.  the  probabilities 
that  these  events  will  occur  when  the  condition  F,  represented 
by  the  logical  equation  V=  1 , is  satisfied,  are 


M [yF]  [zF] 

[F]’  [F]»  "[F]**®” 

in  which  [x  F]  denotes  the  result  obtained  by  multiplying  F by 
x,  according  to  the  rules  of  the  Calculus  of  Logic,  and  changing 
in  the  result  x,  y,  z,  into  p,  q,  r,  &c.  But  the  above  condi- 
tioned probabilities  are  by  hypothesis  equal  to  p,  q,  r,  . . re- 
spectively. Hence  we  have, 


MI 


=p, 


Ml 


= ?» 


MI 

m 


= r,  &c., 


from  which  system  of  equations  equal  in  number  to  the  quanti- 
ties p',  q,  r,  . . , the  values  of  those  quantities  may  be  deter- 
mined. 

Now  sF  consists  simply  of  those  constituents  in  F of  which 
a;  is  a factor.  Let  this  sum  be  represented  by  Vx,  and  in  like 
manner  let  y V be  represented  by  Vy,  &c.  Our  equations  then 
assume  the  form 

mm  ra_,&c.  m 

[F]  [F]  ~ ^ 5 vO 


where  [ FJ  denotes  the  results  obtained  by  changing  in  Vx  the 
symbols  x,  y,  z,  &c.,  into  p',  q',  r,'  &c. 

To  render  the  meaning  of  the  general  problem  and  the  prin- 


262 


GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

ciple  of  its  solution  more  evident,  let  us  take  the  following  ex- 
ample. Suppose  that  in  the  drawing  of  balls  from  an  urn 
attention  had  only  been  paid  to  those  cases  in  which  the  balls 
drawn  were  either  of  a particular  colour,  “white,”  or  of  a par- 
ticular composition,  “ marble,”  or  were  marked  by  both  these 
characters,  no  record  having  been  kept  of  those  cases  in  which  a 
ball  that  was  neither  white  nor  of  marble  had  been  drawn.  Let 
it  then  have  been  found,  that  whenever  the  supposed  condition 
was  satisfied,  there  was  a probability  p that  a white  ball  would  be 
drawn,  and  a probability  q that  a marble  ball  would  be  drawn  : and 
from  these  data  alone  let  it  be  required  to  find  the  probability 
that  in  the  next  drawing,  without  reference  at  all  to  the  condi- 
tion above  mentioned,  a white  ball  will  be  drawn  ; also  the  pro- 
bability that  a marble  ball  will  be  drawn. 

Here  if  x represent  the  drawing  of  a white  ball,  y that  of  a 
marble  ball,  the  condition  V will  be  represented  by  the  logical 
function 

xy  + ®(1  - y)  + (1  - x)  y. 

Hence  we  have 

Vx  = xy  + x (1  - y)  = x,  Vy  = xy  + (1  - x)  y = y ; 
whence 

m=p,  [py-g; 

and  the  final  equations  of  the  problem  are 

P = = . 

p'q'  + p'(i-q')  + q'(i-p)  p’  p'q+p'O-  -q)  + q' 0-  -pr)  q’ 

from  which  we  find 

= p+q~ 1 , = p+q-\ 

p q ’ q P 

It  is  seen  that  p and  q are  respectively  proportional  to  p and 
q,  as  by  Professor  Donkin’s  principle  they  ought  to  be.  The 
solution  of  this  class  of  problems  might  indeed,  by  a direct  appli- 
cation of  that  principle,  be  obtained. 

To  meet  a possible  objection,  I here  remark,  that  the  above 
reasoning  does  not  require  that  the  drawings  of  a white  and  a 
marble  ball  should  be  independent,  in  virtue  of  the  physical  con- 
stitution of  the  balls.  The  assumption  of  their  independence  is 
indeed  involved  in  the  solution,  but  it  does  not  rest  upon  any 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  263 

prior  assumption  as  to  the  nature  of  the  balls,  and  their  relations, 
or  freedom  from  relations,  of  form,  colour,  structure,  &c.  It  is 
founded  upon  our  total  ignorance  of  all  these  things.  Probabi- 
lity always  has  reference  to  the  state  of  our  actual  knowledge, 
and  its  numerical  value  varies  with  varying  information. 

Proposition  III. 

11.  To  determine  in  any  question  of  probabilities  the  logical 
connexion  of  the  quossitum  with  the  data;  that  is,  to  assign  the  event 
whose  probability  is  sought,  as  a logical  function  of  the  event  whose 
probabilities  are  given. 

Let  S,  T,  &c.,  represent  any  compound  events  whose  pro- 
babilities are  given,  S and  T being  in  expression  known  func- 
tions of  the  symbols  x,  y,  z,  &c.,  representing  simple  events. 
Similarly  let  W represent  any  event  whose  probability  is  sought, 
W being  also  a known  function  of  x,  y,  z,  &c.  As  S,  T,  . . W 
must  satisfy  the  fundamental  law  of  duality,  we  are  permitted 
to  replace  them  by  single  logical  symbols,  s,  t,  . . w.  Assume 
then 

s = S,  t - T,  w = W. 

These,  by  the  definition  of  S,  T,  . . W,  will  be  a series  of 
logical  equations  connecting  the  symbols  s,  t,  . . w,  with  the  sym- 
bols x,  y,z  . . 

By  the  methods  of  the  Calculus  of  Logic  we  can  eliminate 
from  the  above  system  any  of  the  symbols  x,  y,  z,  . . , repre- 
senting events  whose  probabilities  are  not  given,  and  determine 
ic  as  a developed  function  of  s,  t,  &c.,  and  of  such  of  the  symbols 
x,  y,  z,  &c.,  if  any  such  there  be,  as  correspond  to  events  whose 
probabilities  are  given.  The  result  will  be  of  the  form 

w = A + 0B  + °-C  + ^D, 

where  A,  B,  C,  and  D comprise  among  them  all  the  possible 
constituents  which  can  be  formed  from  the  symbols  s,  t,  &c.,  i.  e. 
from  all  the  symbols  representing  events  whose  probabilities  are 
given. 

The  above  will  evidently  be  the  complete  expression  of  the 
relation  sought.  For  it  fully  determines  the  event  W,  repre- 


264  GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

seated  by  the  single  symbol  w,  as  a function  or  combination  of 
the  events  similarly  denoted  by  the  symbols  s,  t,  &c.,  and  it  as- 
signs by  the  laws  of  the  Calculus  of  Logic  the  condition 

Z>  = 0, 

as  connecting  the  events  s,  t,  &c.,  among  themselves.  We  may, 
therefore,  by  Principle  vi.,  regard  s,  t,  &c.,  as  simple  events,  of 
which  the  combination  iv,  and  the  condition  with  which  it  is  as- 
sociated D,  are  definitely  determined. 

Uniformity  in  the  logical  processes  of  reduction  being  de- 
sirable, I shall  here  state  the  order  which  will  generally  be  pur- 
sued. 

12.  By  (VIII.  8),  the  primitive  equations  are  reducible  to 
the  forms 

5 (1  - S)  + S (1  — s)  = 0 ; 

f(i-T)+T(l-f)  = 0;  (l) 

w(l-  W)  + W(l  - rv)  = 0 ; 

under  which  they  can  be  added  together  without  impairing  then’ 
significance.  We  can  then  eliminate  the  symbols  x,  y,  z,  either 
separately  or  together.  If  the  latter  course  is  chosen,  it  is  ne- 
cessary, after  adding  together  the  equations  of  the  system,  to 
develop  the  result  with  reference  to  all  the  symbols  to  be  elimi- 
nated, and  equate  to  0 the  product  of  all  the  coefficients  of  the 
constituents  (VII.  9). 

As  w is  the  symbol  whose  expression  is  sought,  we  may  also, 
by  Prop.  hi.  Chap,  ix.,  express  the  result  of  elimination  in  the 
form 

Eio  + E'{\  - w)  = 0. 

E and  E being  successively  determined  by  making  in  the 
general  system  (1),  w = 1 and  w-  0,  and  eliminating  the  symbols 
x,  y,  z,  . . Thus  the  single  equations  from  which  E and  E are 
to  be  respectively  determined  become 

s(l-/S)  + £(l-s)  + *(l-T)  + ..+  1-W=0; 

s(\~S)  + S(\ -s)  + f(l- T)  4 T(l-t)  + W=  0. 

From  these  it  only  remains  to  eliminate  x,  y,  z,  &c.,  and  to  de- 
termine w by  subsequent  development. 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES. 


265 


In  the  process  of  elimination  we  may,  if  needful,  avail  our- 
selves of  the  simplifications  of  Props,  i.  and  11.  Chap.  ix. 

13.  Should  the  data,  beside  informing  us  of  the  probabilities 
of  events,  further  assign  among  them  any  explicit  connexion,  such 
connexion  must  be  logically  expressed,  and  the  equation  or  equa- 
tions thus  formed  be  introduced  into  the  general  system. 


Proposition  IV. 

14.  Given  the  probabilities  of  any  system  of  events  ; to  deter- 
mine by  a general  method  the  consequent  or  derived  probability  of 
any  other  event. 

As  in  the  last  Proposition,  let  S,  T,  &c.,  be  the  events  whose 
probabilities  are  given,  W the  event  whose  probability  is  sought, 
these  being  known  functions  of  x,  y,  z,  &c.  Let  us  represent  the 
data  as  follows : 

Probability  of  & = p ; 

Probability  of  T = q ; ^ ^ 


and  so  on,  p,  q,  &c.,  being  known  numerical  values.  If  then 
we  represent  the  compound  event  S by  s,  T by  t,  and  W by  w, 
we  find  by  the  last  proposition, 

w = A + b B + Q-C +^D-,  (2) 


A , B,  C,  and  D being  functions  of  s,  t,  &c.  Moreover  the  data 
(1)  are  transformed  into 

Prob.  s =p,  Prob.  t = q,  &c.  (3) 

Now  the  equation  (2)  is  resolvable  into  the  system 


w = A + qC 
D = 0, 


} 


(4) 


q being  an  indefinite  class  symbol  (VI.  12).  But  since  by  the 
properties  of  constituents  (V.  Prop,  hi.),  we  have 

A + B+  C+  D = l, 


the  second  equation  of  the  above  system  may  be  expressed  in  the 
form 

A + B + C=  1. 


266  GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

If  we  represent  the  function  A + B + C by  F,  the  system  (4) 
becomes 

w = A + qC;  (5) 

V~  !•  («) 

Let  us  for  a moment  consider  this  result.  Since  F is  the  sum 
of  a series  of  constituents  of  s,  t,  &c.,  it  represents  the  compound 
event  in  which  the  simple  events  involved  are  those  denoted  by 
s , t,  &c.  Hence  (6)  shows  that  the  events  denoted  by  s,  t,  &c., 
.and  whose  probabilities  are  p,  q,  &c.,  have  such  probabilities  not 
as  independent  events , but  as  events  subject  to  a certain  condition 
F.  Equation  (5)  expresses  wasa  similarly  conditioned  combi- 
nation of  the  same  events. 

Now  by  Principle  vi.  the  mode  in  which  this  knowledge  of  the 
connexion  of  events  has  been  obtained  does  not  influence  the  mode 
in  which,  when  obtained,  it  is  to  be  employed.  We  must  reason 
upon  it  as  if  experience  had  presented  to  us  the  events  s,  t,  &c., 
as  simple  events,  free  to  enter  into  every  combination,  but  pos- 
sessing, when  actually  subject  to  the  condition  V,  the  probabili- 
ties p,  q,  &c.,  respectively. 

Let  then  p',  q',  . . , be  the  corresponding  probabilities  of  such 
events,  when  the  restriction  V is  removed.  Then  by  Prop.  ii. 
of  the  present  chapter,  these  quantities  will  be  determined  by  the 
system  of  equations, 

_ p - (7  &c  • (7) 

[F]  -P’  £J7]  ~ <KC- 5 \‘J 

and  by  Prop.  i.  the  probability  of  the  event  w under  the  same 
condition  V will  be 

Prob.  w = 

wherein  Vs  denotes  the  sum  of  those  constituents  in  V of  which  s 
is  a factor,  and  [ FJ  what  that  sum  becomes  when  s,  t,  . . , are 
changed  into  p',  q\  . . , respectively.  The  constant  c represents 
the  probability  of  the  indefinite  event  q;  it  is,  therefore,  arbitrary, 
and  admits  of  any  value  from  0 to  1 . 

Now  it  will  be  observed,  that  the  values  of/)',  q',  &c.,  are  de- 
termined from  (7 ) only  in  order  that  they  may  be  substituted  in 
(8),  so  as  to  render  Prob.  tv  a function  of  known  quantities,  p,  q, 


CHAP.  XVII.]  GENERAL  JETHOB  IN  PROBABILITIES. 


267 


&c.  It  is  obvious,  therefore,  that  instead  of  the  letters  pi,  q,  &c., 
we  might  employ  any  others  as  s,  t,  &c.,  in  the  same  quantitative 
acceptations.  This  particular  step  would  simply  involve  a change 
of  meaning  of  the  symbols  s,  t,  &c. — their  ceasing  to  be  logical, 
and  becoming  quantitative.  The  systems  (7)  and  (8)  would  then 
become 

F,  Vt 


y~P’  y ~ V 


Prob.  w = 


&c. 
A + cC 


(9) 

(10) 


In  employing  these,  it  is  only  necessary  to  determine  from  (9) 
s,  t,  &c.,  regarded  as  quantitative  symbols,  in  terms  of p,  q,  &c., 
and  substitute  the  resulting  values  in  (10).  It  is  evident,  that 
s,  t,  &c.,  inasmuch  as  they  represent  probabilities,  will  be  positive 
proper  fractions. 

The  system  (9)  may  be  more  symmetrically  expressed  in  the 
form 

V V, 

— = — F.  (11) 

p q 

Or  we  may  express  both  (9)  and  (10)  together  in  the  symme- 
trical system 

= = ti±2C  = F;  (12) 

p q u v ' 

wherein  u represents  Prob.  iv. 

15.  It  remains  to  interpret  the  constant  c assumed  to  repre- 
sent the  probability  of  the  indefinite  event  q.  Now  the  logical 
equation 

w = A + qC, 

interpreted  in  the  reverse  order,  implies  that  if  either  the  event 
A take  place,  or  the  event  C in  connexion  with  the  event  q,  the 
event  w will  take  place,  and  not  otherwise.  Hence  q represents 
that  condition  under  which,  if  the  event  C take  place,  the  event 
w will  take  place.  But  the  probability  of  q is  c.  Hence,  there- 
fore, c = probability  that  if  the  event  C take  place  the  event  w 
will  take  place. 

Wherefore  by  Principle  ii., 

Probability  of  concurrence  of  C and  w 


c = 


Probability  of  C 


268 


GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 


We  may  hence  determine  the  nature  of  that  new  experience 
from  which  the  actual  value  of  c may  be  obtained.  For  if  we 
substitute  in  C for  s,  t,  &c.,  their  original  expressions  as  func- 
tions of  the  simple  events  x,  y,  z,  &c.,  we  shall  form  the  ex- 
pression of  that  event  whose  probability  constitutes  the  denomi- 
nator of  the  above  value  of  c ; and  if  we  multiply  that  expression 
by  the  original  expression  of  w,  we  shall  form  the  expression  of 
that  event  whose  probability  constitutes  the  numerator  of  c,  and 
the  ratio  of  the  frequency  of  this  event  to  that  of  the  former  one,  de- 
termined by  new  observations,  will  give  the  value  of  c.  Let  it  be 
remarked  here,  that  the  constant  c does  not  necessarily  make  its 
appearance  in  the  solution  of  a problem.  It  is  only  when  the 
data  are  insufficient  to  render  determinate  the  probability  sought, 
that  this  arbitrary  element  presents  itself,  and  in  this  case  it  is 
seen  that  the  final  logical  equation  (2)  or  (5)  informs  us  how  it 
is  to  be  determined. 

If  that  new  experience  by  which  c may  be  determined  can- 
not be  obtained,  we  can  still,  by  assigning  to  c its  limiting  values 
0 and  1,  determine  the  limits  of  the  probability  of  w.  These 
are 

A 

Minor  limit  of  Prob.  w = 


Superior  limit 


A + C 


Between  these  limits,  it  is  certain  that  the  probability  sought 
must  lie  independently  of  all  new  experience  which  does  not  ab- 
solutely contradict  the  past. 

If  the  expression  of  the  event  C consists  of  many  constituents, 
the  logical  value  of  w being  of  the  form 

w = A + — C\  + — Ci  + &c., 


we  can,  instead  of  employing  their  aggregate  as  above,  present 
the  final  solution  in  the  form 

T . , A 4*  C\C/\  4*  CiC<i  + &c. 

Prob.  w = 


V 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  269 

Here  cx  = probability  that  if  the  event  Cx  occur,  the  event  w will 
occur,  and  so  on  for  the  others.  Convenience  must  decide  which 
form  is  to  be  preferred. 

16.  The  above  is  the  complete  theoretical  solution  of  the 
problem  proposed.  It  may  be  added,  that  it  is  applicable  equally 
to  the  case  in  which  any  of  the  events  mentioned  in  its  original 
statement  are  conditioned.  Thus,  if  one  of  the  data  is  the  proba- 
bility p,  that  if  the  event  x occur  the  event  y will  occur ; the 
probability  of  the  occurrence  of  x not  being  given,  we  must  as- 
sume Prob.  x = c (an  arbitrary  constant),  then  Prob.  xy  = cp,  and 
these  two  conditions  must  be  introduced  into  the  data,  and  em- 
ployed according  to  the  previous  method.  Again,  if  it  is  sought 
to  determine  the  probability  that  if  an  event  x occur  an  event  y 
will  occur,  the  solution  will  assume  the  form 

Prob.  sought  = — , 

6 Prob.  x ’ 

the  numerator  and  denominator  of  which  must  be  separately  de- 
termined by  the  previous  general  method. 

17.  We  are  enabled  by  the  results  of  these  investigations  to 
establish  a general  rule  for  the  solution  of  questions  in  probabi- 
lities. 


General  Rule. 

Case  I. — When  all  the  events  are  unconditioned. 

Form  the  symbolical  expressions  of  the  events  whose  proba- 
bilities are  given  or  sought. 

Equate  such  of  those  expressions  as  relate  to  compound  events 
to  a new  series  of  symbols,  s,  t,  &c.,  which  symbols  regard  as  re- 
presenting the  events,  no  longer  as  compound  but  simple,  to 
whose  expressions  they  have  been  equated. 

Eliminate  from  the  equations  thus  formed  all  the  logical  sym- 
bols, except  those  which  express  events,  s,  t,  &c.,  whose  respective 
probabilities  p,  q,  &c.  are  given,  or  the  event  w whose  probability 
is  sought,  and  determine  w as  a developed  function  of  s,  t,  &c. 
in  the  form 

w = A + OH  + - C + ^ D. 


270  GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 


Let  A + B+  C = V,  and  let  Vs  represent  the  aggregate  of 
those  constituents  in  V which  contain  s as  a factor,  Vt  of  those 
which  contain  t as  a factor,  and  thus  for  all  the  symbols  whose 
probabilities  are  given. 

Then,  passing  from  Logic  to  Algebra,  form  the  equations 


Prob.  to  = 


A + cC 
~V~’ 


(1) 

(2) 


from  (1)  determine  s,  t,  &c.  as  functions  of  p,  q,  &c.,  and  sub- 
stitute their  values  in  (2).  The  result  will  express  the  solution 
required. 

Or  form  the  symmetrical  system  of  equations 

Vs_Vt  _A  + cC_  V 
p q “ U 1’  W 

where  u represents  the  probability  sought. 

If  c appear  in  the  solution,  its  interpretation  will  be 

Prob.  Cw 
Prob. c ’ 

and  this  interpretation  indicates  the  nature  of  the  experience 
which  is  necessary  for  its  discovery. 

Case  II. — When  some  of  the  events  are  conditioned. 

If  there  be  given  the  probability  p that  if  the  event  X occur, 
the  event  Y will  occur,  and  if  the  probability  of  the  antecedent 
X be  not  given,  resolve  the  proposition  into  the  two  following, 
viz. : 

Probability  of  X = c, 

Probability  of  AT  = cp. 

If  the  quacsitum  be  the  probability  that  if  the  event  W occur, 
the  event  Z will  occur,  determine  separately,  by  the  previous 
case,  the  terms  of  the  fraction 

Prob.  WZ 
Prob.  W ’ 

and  the  fraction  itself  will  express  the  probability  sought. 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  271 

It  is  understood  in  this  case  that  X,  Y,  W,  Z may  be  any 
compound  events  whatsoever.  The  expressions  XY  and  WZ 
represent  the  products  of  the  symbolical  expressions  of  X and  Y 
and  of  W and  Z,  formed  according  to  the  rules  of  the  Calculus  of 
Logic. 

The  determination  of  the  single  constant  c may  in  certain 
cases  be  resolved  into,  or  replaced  by,  the  determination  of  a series 
of  arbitrary  constants  clt  c2 . . according  to  convenience,  as  pre- 
viously explained. 

18.  It  has  been  stated  (I.  12)  that  there  exist  two  distinct  de- 
finitions, or  modes  of  conception,  upon  which  the  theory  of  pro- 
babilities may  be  made  to  depend,  one  of  them  being  connected 
more  immediately  with  Number,  the  other  more  directly  with 
Logic.  W e have  now  considered  the  consequences  which  flow 
from  the  numerical  definition,  and  have  shown  how  it  conducts 
us  to  a point  in  which  the  necessity  of  a connexion  with  Logic 
obviously  suggests  itself.  We  have  seen  to  some  extent  what 
is  the  nature  of  that  connexion ; and  further,  in  what  manner  the 
peculiar  processes  of  Logic,  and  the  more  familiar  ones  of  quanti- 
tative Algebra,  are  involved  in  the  same  general  method  of  solu- 
tion, each  of  these  so  accomplishing  its  own  object  that  the  two 
processes  may  be  regarded  as  supplementary  to  each  other.  It 
remains  to  institute  the  reverse  order  of  investigation,  and,  setting 
out  from  a definition  of  probability  in  which  the  logical  relation 
is  more  immediately  involved,  to  show  how  the  numerical  defini- 
tion would  thence  arise,  and  how  the  same  general  method, 
equally  dependent  upon  both  elements,  would  finally,  but  by  a 
different  order  of  procedure,  be  established. 

That  between  the  symbolical  expressions  of  the  logical  cal- 
culus and  those  of  Algebra  there  exists  a close  analogy,  is  a fact 
to  which  attention  has  frequently  been  directed  iii  the  course  of 
the  present  treatise.  It  might  even  be  said  that  they  possess  a 
community  of  forms,  and,  to  a very  considerable  degree,  a com- 
munity of  laws.  With  a single  exception  in  the  latter  respect, 
their  difference  is  only  one  of  interpretation.  Thus  the  same 
expression  admits  of  a logical  or  of  a quantitative  interpretation, 
according  to  the  particular  meaning  which  we  attach  to  the  sym- 


272  GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

bols  it  involves.  The  expression  xy  represents,  under  the  former 
condition,  a concurrence  of  the  events  denoted  by  x and  y ; under 
the  latter,  the  product  of  the  numbers  or  quantities  denoted  by  x 
and  y.  And  thus  every  expression  denoting  an  event,  simple  or 
compound,  admits,  under  another  system  of  interpretation,  of  a 
meaning  purely  quantitative.  Here  then  arises  the  question, 
whether  there  exists  any  principle  of  transition,  in  accordance 
with  which  the  logical  and  the  numerical  interpretations  of  the 
same  symbolical  expression  shall  have  an  intelligible  connexion. 
And  to  this  question  the  following  considerations  afford  an 
answer. 

19.  Let  it  be  granted  that  there  exists  such  a feeling  as  ex- 
pectation, a feeling  of  which  the  object  is  the  occurrence  of  events, 
and  which  admits  of  differing  degrees  of  intensity.  Let  it  also 
be  granted  that  this  feeling  of  expectation  accompanies  our 
knowledge  of  the  circumstances  under  which  events  are  produced, 
and  that  it  varies  with  the  degree  and  kind  of  that  knowledge. 
Then,  without  assuming,  or  tacitly  implying,  that  the  intensity 
of  the  feeling  of  expectation,  viewed  as  a mental  emotion,  admits 
of  precise  numerical  measurement,  it  is  perfectly  legitimate  to 
inquire  into  the  possibility  of  a mode  of  numerical  estimation 
which  shall,  at  least,  satisfy  these  following  conditions,  viz.,  that 
the  numerical  value  which  it  assigns  shall  increase  when  the 
known  circumstances  of  an  event  are  felt  to  justify  a stronger 
expectation,  shall  diminish  when  they  demand  a weaker  expec- 
tation, and  shall  remain  constant  when  they  obviously  require  an 
equal  degree  of  expectation. 

Now  these  conditions  at  least  will  be  satisfied,  if  we  assume 
the  fundamental  principle  of  expectation  to  be  this,  viz.,  that  the 
laws  for  the  expression  of  expectation,  viewed  as  a numerical 
element,  shall  be  the  same  as  the  laws  for  the  expression  of  the 
expected  event  viewed  as  a logical  element.  Thus  if  <p  (x,  y,  z)  re- 
present any  unconditional  event  compounded  in  any  manner  of 
the  events  x,  y,  2,  let  the  same  expression  <p  (x,  y,  2),  according 
to  the  above  principle,  denote  the  expectation  of  that  event; 
x,  y,  2 representing  no  longer  the  simple  events  involved,  but 
the  expectations  of  those  events. 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  273 

For,  in  the  first  place,  it  is  evident  that,  under  this  hypothesis, 
the  probability  of  the  occurrence  of  some  one  of  a set  of  mutually 
exclusive  events  will  be  equal  to  the  sum  of  the  separate  proba- 
bilities of  those  events.  Thus  if  the  alternation  in  question  con- 
sist of  n mutually  exclusive  events  whose  expressions  are 

0i(*»  */>  z),  tf>2  0,  y,z), 0„  (x,  y,  z), 

the  expression  of  that  alternation  will  be 

01  (*,  y , Z)  + 02  (x,  y,z)  . . + 0n  (x,  y,  z)  = 1 ; 

the  literal  symbols  x,  y,  z being  logical,  and  relating  to  the  sim- 
ple events  of  which  the  three  alternatives  are  compounded : 
and,  by  hypothesis,  the  expression  of  the  probability  that  some 
one  of  those  alternatives  will  occur  is 

0i  (*,  V,  z ) + 02  (®,  y,z)..  + $n  0,  y,  z)t 

x,  y , z here  denoting  the  probabilities  of  the  above  simple  events. 
Now  this  expression  increases,  cceteris  paribus , with  the  increase 
of  the  number  of  the  alternatives  which  are  involved,  and  di- 
minishes with  the  diminution  of  their  number ; which  is  agree- 
able to  the  condition  stated. 

Furthermore,  if  we  set  out  from  the  above  hypothetical  defi- 
nition of  the  measure  of  probability,  we  shall  be  conducted, 
either  by  necessary  inference  or  by  successive  steps  of  suggestion, 
which  might  perhaps  be  termed  necessary , to  the  received  nu- 
merical definition.  We  are  at  once  led  to  recognise  unity  (1) 
as  the  proper  numerical  measure  of  certainty.  For  it  is  certain 
that  any  event  x or  its  contrary  1 - x will  occur.  The  expres- 
sion of  this  proposition  is 

x + (1  - x}  = 1, 

whence,  by  hypothesis,  x+  (1  - x),  the  measure  of  the  proba- 
bility of  the  above  proposition,  becomes  the  measure  of  certainty. 
But  the  value  of  that  expression  is  1,  whatever  the  particular 
value  of  x may  be.  Unity,  or  1,  is  therefore,  on  the  hypothesis 
in  question,  the  measure  of  certainty. 

Let  there,  in  the  next  place,  be  n mutually  exclusive,  but 
equally  possible  events,  which  we  will  represent  by  <15  t2,  . . tn. 


274 


GENERAL  METHOD  IN  PROBABILITIES.  [CHAP.  XVII. 

The  proposition  which  affirms  that  some  one  of  these  must  occur 
will  be  expressed  by  the  equation 

#1  + t2  . . + tn  — 1 } 

and,  as  when  we  pass  in  accordance  with  the  reasoning  of  the 
last  section  to  numerical  probabilities,  the  same  equation  remains 
true  in  form,  and  as  the  probabilities  tx,  t2. . t„  are  equal,  we 
have 

ntx  = 1, 

1 11 

whence  tx  = -,  and  similarly  t2  ■=-,£„  = -.  Suppose  it  then  re- 
quired to  determine  the  probability  that  some  one  event  of  the 
partial  series  tx,  t2 . . tm  will  occur,  we  have  for  the  expression 
required 

tx  + t.2 . . + tm  = - + to  m terms 
n n 

m 

n 

Hence,  therefore,  if  there  are  m cases  favourable  to  the  occur- 
rence of  a particular  alternation  of  events  out  of  n possible  and 
equally  probable  cases,  the  probability  of  the  occurrence  of  that 

JYI 

alternation  will  be  expressed  by  the  fraction  — . 

Now  the  occurrence  of  any  event  which  may  happen  in  diffe- 
rent equally  possible  ways  is  really  equivalent  to  the  occurrence 
of  an  alternation,  i.  e.,  of  some  one  out  of  a set  of  alternatives. 
Hence  the  probability  of  the  occurrence  of  any  event  may  be 
expressed  by  a fraction  whose  numerator  represents  the  number 
of  cases  favourable  to  its  occurrence,  and  denominator  the  total 
number  of  equally  possible  cases.  But  this  is  the  rigorous  nume- 
rical definition  of  the  measure  of  probability.  That  definition  is 
therefore  involved  in  the  more  peculiarly  logical  definition,  the 
consequences  of  which  we  have  endeavoured  to  trace. 

20.  From  the  above  investigations  it  clearly  appears,  1st, 
that  whether  we  set  out  from  the  ordinary  numerical  definition 
of  the  measure  of  probability,  or  from  the  definition  which  assigns 
to  the  numerical  measure  of  probability  such  a law  of  value  as 
shall  establish  a formal  identity  between  the  logical  expressions 


CHAP.  XVII.]  GENERAL  METHOD  IN  PROBABILITIES.  275 

of  events  and  the  algebraic  expressions  of  their  values,  we  shall 
be  led  to  the  same  system  of  practical  results.  2ndly,  that 
either  of  these  definitions  pursued  to  its  consequences,  and  con- 
sidered in  connexion  with  the  relations  which  it  inseparably  in- 
volves, conducts  us,  by  inference  or  suggestion,  to  the  other 
definition.  To  a scientific  view  of  the  theory  of  probabilities 
it  is  essential  that  both  principles  should  be  viewed  together,  in 
their  mutual  bearing  and  dependence. 


276 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 


CHAPTER  XVIII. 

ELEMENTARY  ILLUSTRATIONS  OF  THE  GENERAL  METHOD  IN  PROBA- 
BILITIES. 

1.  TT  is  designed  here  to  illustrate,  by  elementary  examples, 
the  general  method  demonstrated  in  the  last  chapter. 
The  examples  chosen  will  be  chiefly  such  as,  from  their  sim- 
plicity, permit  a ready  verification  of  the  solutions  obtained. 
But  some  intimations  will  appear  of  a higher  class  of  problems, 
hereafter  to  be  more  fully  considered,  the  analysis  of  which 
would  be  incomplete  without  the  aid  of  a distinct  method  deter- 
mining the  necessary  conditions  among  their  data,  in  order  that 
they  may  represent  a possible  experience,  and  assigning  the  cor- 
responding limits  of  the  final  solutions.  The  fuller  consideration 
of  that  method,  and  of  its  applications,  is  reserved  for  the  next 
chapter. 

2.  Ex.  1. — The  probability  that  it  thunders  upon  a given 
day  is  p,  the  probability  that  it  both  thunders  and  hails  is  q,  but 
of  the  connexion  of  the  two  phenomena  of  thunder  and  hail,  no- 
thing further  is  supposed  to  be  known.  Required  the  probability 
that  it  hails  on  the  proposed  day. 

Let  x represent  the  event — It  thunders. 

Let  y represent  the  event — It  hails. 

Then  xy  will  represent  the  event — It  thunders  and  hails ; and 
the  data  of  the  problem  are 

Prob.  x = p,  Prob.  xy  = q. 

There  being  here  but  one  compound  event  xy  involved,  assume, 
according  to  the  rule, 

xy  = -u.  (1) 

Our  data  then  become 

Prob.  x = p,  Prob.  w = q 
and  it  is  required  to  find  Prob.  y.  Now  (1)  gives 


(2) 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS.  277 

y = - = w;z  + ^w(l-a;)  + 0 (1  - w)  a;  + (1  - w)  (1  - #). 

Hence  (XVII.  17)  we  find 

V = ux  + (1  - u)  x + (1  - u)  (1  -x), 

Vx  = ux  + (1  - u)  x = x , Vu  = ux ; 
and  the  equations  of  the  General  Rule,  viz., 

P 9 


Prob. y = 


A + cC 

v~ 


become,  on  substitution,  and  observing  that  A = ux,  C=  (1  - u) 
(1  - x),  and  that  V reduces  to  x + (1  - u)  (1  - x), 


x ux  . 

_ = _ = z + ( 1 - u)  (1  - x), 
p q ' ' 


Prob. y 


ux  + c ( 1 - u)  ( 1 - x) 
X 4 (1  - u)  (1  - x)  ’ 


from  which  we  readily  deduce,  by  elimination  of  x and  u, 
Prob.  y - q + c (1  - p). 


(3) 

(4) 

(5) 


In  this  result  c represents  the  unknown  probability  that  if  the 
event  (1  -u)  (1  - x)  happen,  the  event  y will  happen.  Now 
(l-«)(l-a;)  = (l  - xy)  (1  - x)  = 1 - x,  on  actual  multiplication. 
Hence  c is  the  unknown  probability  that  if  it  do  not  thunder,  it 
will  hail. 

The  general  solution  (5)  may  therefore  be  interpreted  as  fol- 
lows : — The  probability  that  it  hails  is  equal  to  the  probability 
that  it  thunders  and  hails,  q,  together  with  the  probability  that  it 
does  not  thunder,  1 -p,  multiplied  by  the  probability  c,  that  if  it 
does  not  thunder  it  will  hail.  And  common  reasoning  verifies 
this  result. 

If  c cannot  be  numerically  determined,  we  find,  on  assigning 
to  it  the  limiting  values  0 and  1,  the  following  limits  of  Prob.  y, 
viz. : 

Inferior  limit  = q. 

Superior  limit  -q+  1 - p. 


278  ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 

3.  Ex.  2. — The  probability  that  one  or  both  of  two  events 
happen  is  p,  that  one  or  both  of  them  fail  is  q.  What  is  the 
probability  that  only  one  of  these  happens  ? 

Let  x and  y represent  the  respective  events,  then  the  data 
are — 

Prob.  xy  + x (1  -y)  + (1  - x)  y = p, 

Prob.  x(l  - y)  + (1  -x)y  + (1  - x)  (1  - y)  = q; 
and  we  are  to  find 

Prob.  a:  (1  - y)  + y (l  - x). 

Here  all  the  events  concerned  being  compound,  assume 

xy  + x (1  - y)  + (1  -x)y  = s, 
x{\-y)  + (1  ~x)y  + (1  -x)(l  - y ) = t, 
r(l  -,y)  + (l  - x)  y = w. 

Then  eliminating  x and  y,  and  determining  w as  a developed 
function  of  s and  t,  we  find 

w = st  + 0 5(1  - 1)  + 0 (1  - s)t  + ^ (1  - 5)  (1  - 1). 

Hence  A = st,  C=  0,  V=  st  + s (1  - t)  + (1  - s)  t = s + (1  - s)t, 
Vs^s,  Vt  <=t;  and  the  equations  of  the  General  Rule  (XVII.  17) 
become 

(1) 

pq 

Prob.  w = ; 

s + (1  - s)t 

whence  we  find,  on  eliminating  s and  t, 

Prob.  w = p + q - 1 . 

Hence  p + q - 1 is  the  measure  of  the  probability  sought.  This 
result  may  be  verified  as  follows  : — Since  p is  the  probability  that 
one  or  both  of  the  given  events  occur,  1 - p will  be  the  proba- 
bility that  they  both  fail ; and  since  q is  the  probability  that  one 
or  both  fail,  1 - q is  the  probability  that  they  both  happen. 
Hence  1 - p + 1 - q,  or  2 -p  - q,  is  the  probability  that  they 
either  both  happen  or  both  fail.  But  the  only  remaining  alter- 
native which  is  possible  is  that  one  alone  of  the  events  happens. 
Hence  the  probability  of  this  occurrence  is  1 - (2  - p - q),  or 
p + q - 1,  as  above. 


ELEMENTARY  ILLUSTRATIONS. 


279 


CHAP.  XVIII.] 

4.  Ex.  3. — The  probability  that  a witness  A speaks  the  truth 
is  p,  the  probability  that  another  witness  B speaks  the  truth  is  q, 
and  the  probability  that  they  disagree  in  a statement  is  r.  What 
is  the  probability  that  if  they  agree,  their  statement  is  true  ? 

Let  x represent  the  hypothesis  that  A speaks  truth ; y that 
B speaks  truth ; then  the  hypothesis  that  A and  B disagree  in 
their  statement  will  be  represented  byar(l  - y)  + y(\~x)\  the 
hypothesis  that  they  agree  in  statement  by  xy  + (1  - x)  (1  - y), 
and  the  hypothesis  that  they  agree  in  the  truth  by  xy.  Hence 
we  have  the  following  data  : 

Prob.  x = p,  Prob -y  = q,  Prob.  x (1  - y)  + y (1  - x)  = r, 
from  which  we  are  to  determine 

Prob .xy 

Prob.  xy  + (1  - x)  (1  - y)' 

But  as  Prob.  x (1  - y)  + y (1  - x)  = r,  it  is  evident  that  Prob. 
xy  + (1  - x)  (1  - y)  will  be  1 - r ; we  have  therefore  to  seek 

Prob. xy 
1 - r 

Now  the  compound  events  concerned  being  in  expression, 
x (1  - y)  + y (1  - x)  and  xy,  let  us  assume 

x(l  -y)  + y(l-x)  = s 1 

xy  = w J ' 

Our  data  then  are  Prob.  x = p,  Prob.  y = q,  Prob.  s = r,  and  we 
are  to  find  Prob.  w. 

The  system  (1)  gives,  on  reduction, 

\x{\-y)+y{\-x))  (l-*)  + j{ay+(l-a;)  (1-2/)} 

+ xy  (1  - w)  + w (1  - xy)  = 0 ; 

whence 

ftp  -y)(!  -s)  +y(l-a)  O -j)  + «cy  + j(l  - a;)  (1  -y)+  xy 

2 xy  - 1 

= \XVS  + xy(\-s)  +(h;(l  - y)  s + ^ x (1  - y)  (1  - s) 

1 1 

+ 0(l-*)y»+-(l-®)(l— y)*  + -(l-a)  y(l-s) 

+ 0(1-*)  (l-y)(l  -s). 


(2) 


280 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 


In  the  expression  of  this  development,  the  coefficient  - has  been 
made  to  replace  every  equivalent  form  (X.  6).  Here  we  have 
V = xy  (1  - s)  + x (1  - y)  s + (1-  x)  ys  + (1  - x)  (1  - y)  (1  - s')  ; 
whence,  passing  from  Logic  to  Algebra, 
xy  (1  - s)  + a;(l  - y)  s _ xy  (1  - s)  + (1  -x)ys 
~P  ” ' 9 

a?  (1  - y)  s + (1  - x)  ys 
r 

= xy{\  -s)  +«(1  -y)  s + (1  - x)ys  + (1  - a)  (1  - y)  (1  - s). 


Prob. w 


ay  (1  - s ) 

-y)«+(l-a)  ys  + (l-ar)(l-y)(l-«)* 


from  which  we  readily  deduce 


td  u p + q - r 
Prob.  w = - — ; 

Li 

whence  we  have 

Prob.  xy  p + q - r 
1-r  = 2 ( 1 - r)  ( ' 

for  the  value  sought. 

If  in  the  same  way  we  seek  the  probability  that  if  A and  B 
agree  in  their  statement,  that  statement  will  be  false,  we  must 
replace  the  second  equation  of  the  system  (1)  by  the  following, 
viz. : 

(1  - x)  (1  - y)  = w ; 
the  final  logical  equation  will  then  be 

1 1 

w = - xys  + Oxy  (1  - s)  + Ox  (1  - y)  s + ^x(l-y)  (1-s) 

+ 0 (1  - x)  ys  + ^ (1  -x)y  (1  - s)  + i (1  - x)  (1  - y)s 

+ (1  -*)  (i  -y)  C1  - s)’  (4) 

whence,  proceeding  as  before,  we  finally  deduce 

Prob  ,wAzJL-<LlL.  (5) 

A 


Wherefore  we  have 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS. 

Prob.  (1  - x)  (1  - y)  2 - p - q - r 
1 - r 2 (1  - r) 

for  the  value  here  sought. 

These  results  are  mutually  consistent.  For  since  it  is  certain 
that  the  joint  statement  of  A and  B must  be  either  true  or  false, 
the  second  members  of  (3)  and  (5)  ought  by  addition  to  make  1. 
Now  we  have  identically, 

p + q - r 2 - p - q - r 
2(1  - r)  + 2(1  -T)  ' 

It  is  probable,  from  the  simplicity  of  the  results  (5)  and  (6), 
that  they  might  easily  be  deduced  by  the  application  of  known 
principles ; but  it  is  to  be  remarked  that  they  do  not  fall  directly 
within  the  scope  of  known  methods.  The  number  of  the  data 
exceeds  that  of  the  simple  events  which  they  involve.  M.  Cour- 
not, in  his  very  able  work,  “ Exposition  de  la  Theorie  des 
Chances,”  has  proposed,  in  such  cases  as  the  above,  to  select 
from  the  original  premises  different  sets  of  data,  each  set  equal  in 
number  to  the  simple  events  which  they  involve,  to  assume  that 
those  simple  events  are  independent,  determine  separately  from 
the  respective  sets  of  the  data  their  probabilities,  and  comparing 
the  different  values  thus  found  for  the  same  elements,  judge  how 
far  the  assumption  of  independence  is  justified.  This  method 
can  only  approach  to  correctness  when  the  said  simple  events 
prove,  according  to  the  above  criterion,  to  be  nearly  or  quite  in- 
dependent ; and  in  the  questions  of  testimony  and  of  judgment, 
in  which  such  an  hypothesis  is  adopted,  it  seems  doubtful  whether 
it  is  justified  by  actual  experience  of  the  ways  of  men. 

5.  Ex.  4. — From  observations  made  during  a period  of  gene- 
ral sickness,  there  was  a probability  p that  any  house  taken  at 
random  in  a particular  district  was  visited  by  fever,  a probability 
q that  it  was  visited  by  cholera,  and  a probability  r that  it  es- 
caped both  diseases,  and  was  not  in  a defective  sanitary  condition 
as  regarded  cleanliness  and  ventilation.  What  is  the  probability 
that  any  house  taken  at  random  was  in  a defective  sanitary 
condition  ? 

With  reference  to  any  house,  let  us  appropriate  the  symbols 
x,  y,  z,  as  follows,  viz. : 


281 

(6) 


282 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 

The  symbol  x to  the  visitation  of  fever. 

y „ cholera. 

z defective  sanitary  condition. 

The  events  whose  probabilities  are  given  are  then  denoted  by 
x,  y,  and  (l  - a:)  (1  - y)  (1  - z),  the  event  whose  probability  is 
sought  is  z.  Assume  then, 

(l-a:)(l-y)(l-z)  = u>; 

then  our  data  are, 

Prob.  x = p,  Prob.  y = q,  Prob.  w = r, 
and  we  are  to  find  Prob.  z.  Now 

(1  — aQO  -y)-  m 

(l-x)(l-y) 

= \*yw  + ~w)+\x  (l-y)to  + £ x(l-y)  (1  - u>) 

+ 5 (l-*)yw  + ^(l-®)y(l  -w)  + 0 (l-*)(l-y)  to 

+ (i-*)  0-y)  (i-®)-  0) 

The  value  of  V deduced  from  the  above  is 

V=  xy  (1  - w)  + x (1  - y)  (1  - w)  + (1  — x)  y (1  — w) 

+ (l  - x)  (l  - y)  w + (l  - x)  (l  - y)  (1-  w)  = l - w + w (l  - x)  (l  - y) ; 
and  similarly  reducing  Vx,  Vy,  Vw,  we  get 

Vx  = x (1  - to),  vy  = y (1  - to),  Vw  = w( l-x)  (1  - y)  ; 
furnishing  the  algebraic  equations 

x(l-w)  y(\  -w)  w(\-x)(\  -y)  , . ,,  . 

-X— =^q — ~ = ~ r — = 1-W,  + M,(1“a;)(1“y)-  (2) 

As  respects  those  terms  of  the  development  characterized  by 
the  coefficients  -,  I shall,  instead  of  collecting  them  into  a single 

term,  present  them,  for  the  sake  of  variety  (XVII.  18),  in  the 
form 

jj  x(\  - io)+  ^(1  -x)y{\  - to); 
the  value  of  Prob.  z will  then  be 


(3) 


283 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS. 

Prob. z = C1-^)  (l-y)(l-to)  + cg(l-io)  + cf(l-a!)y  (l-io) 

1 - w + w ( 1 - Xs)  ( 1 - y) 

From  (2)  and  (4)  we  deduce 

-d  i.  (1  — />  — r)(l— — »*)  ,?(1  -p-r) 

Prob.  z = - — - - + cp  + c — ■ ■■  

1 — r 1 — r 

as  the  expression  of  the  probability  required.  If  in  this  result 
we  make  c = 0,  and  c = 0,  we  find  for  an  inferior  limit  of  its  value 

(\  — P — V)  (\  — Q — 

- - — - ; and  if  we  make  c = 1,  c = 1,  we  obtain 

1 - r 

for  its  superior  limit  1 - r. 

6.  It  appears  from  inspection  of  this  solution,  that  the  pre- 
mises chosen  were  exceedingly  defective.  The  constants  c and 
c indicate  this,  and  the  corresponding  terms  (3)  of  the  final 
logical  equation  show  how  the  deficiency  is  to  be  supplied. 
Thus,  since 

x(l  - w)  = x (1  - (1  - x)  (1  - y)  (1  - z)}  <=  x, 

we  learn  that  c is  the  probability  that  if  any  house  was  visited  by 
fever  its  sanitary  condition  is  defective,  and  that  c is  the  proba- 
bility that  if  any  house  Avas  visited  by  cholera  without  fever,  its 
sanitary  condition  was  defective. 

If  the  terms  of  the  logical  development  affected  by  the  coeffi- 
cient had  been  collected  together  as  in  the  direct  statement  of 

the  general  rule,  the  final  solution  would  have  assumed  the  fol- 
lowing form : 

Prob.  z = + c(p  + q- 

c here  representing  the  probability  that  if  a house  was  visited  by 
either  or  both  of  the  diseases  mentioned,  its  sanitary  condition 
was  defective.  This  result  is  perfectly  consistent  Avith  the  former 
one,  and  indeed  the  necessary  equivalence  of  the  different  forms 
of  solution  presented  in  such  cases  may  be  formally  established. 

The  above  solution  may  be  verified  in  particular  cases.  Thus, 
taking  the  second  form,  if  c = 1 we  find  Prob.  z = 1 - r,  a correct 
result.  For  if  the  presence  of  either  fever  or  cholera  certainly 


284 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 

indicated  a defective  sanitary  condition,  the  probability  that  any 
house  would  be  in  a defective  sanitary  state  would  be  simply 
equal  to  the  probability  that  it  was  not  found  in  that  category 
denoted  by  z,  the  probability  of  which  would,  by  the  data,  be  1 - r. 
Perhaps  the  general  verification  of  the  above  solution  would  be 
difficult. 

The  constants  p,  q,  and  r in  the  above  solution  are  subject  to 
the  conditions 

P + r<  1,  q + r<\. 

7.  Ex.  5. — Given  the  probabilities  of  the  premises  of  a hypo- 
thetical syllogism  to  find  the  probability  of  the  conclusion. 

Let  the  syllogism  in  its  naked  form  be  as  follows  : 

Major  premiss  : If  the  proposition  Y is  true  X is  true. 
Minor  premiss : If  the  proposition  Z is  true  Y is  true. 
Conclusion  : If  the  proposition  Z is  true  X is  true. 

Suppose  the  probability  of  the  major  premiss  to  be  p,  that  of  the 
minor  premiss  q. 

The  data  then  are  as  follows,  representing  the  proposition  X 
by  x,  &c.,  and  assuming  c and  c as  arbitrary  constants  : 

Prob.  y = c,  Prob.  xy  = cp; 

Prob.  2 = c',  Prob.  yz  = c'q ; 
from  which  we  are  to  determine, 

Prob.  xz  Prob.  xz 
Prob.  z °r  c 

Let  us  assume, 

xy  = u,  yz  = v,  xz  = w ; 

then,  proceeding  according  to  the  usual  method  to  determine  w 
as  a developed  function  of  y,  z,  u,  and  v,  the  symbols  corres- 
ponding to  propositions  whose  probabilities  are  given,  we  find 

w = uzvy  + Om  (1  - z)  (1  - v)  y + 0 (1  - u)  zvy 

+ ^(1  -u)z{\ -«)(1  -y)+  0 (1  -u)  (1  -z)  (1  -v)y 

+ 0 (1  - u)  (1  - z)  (1  - v)  ( 1 - y)  + terms  whose  coeffi- 
1 

cients  are  - ; 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS.  285 

and  passing  from  Logic  to  Algebra, 

uzvy  + u (l-*)0-r)  y _uzvy  + (1  - u)zvy  + (1-  u)  z (1  - u)(l- V) 
cp  d 

uzvy  + ( 1 - u)  zvy 
Tq 

uzvy  + u (l  - z)  (l  - v) y + - u)  zvy  + (l-u)(l-z)  (l-v)y_y 

c 


Prob.  to  - , 

wherein 

V = uzvy  + m(1-z)(1  -v)y  + (\  - u ) zvy  + (1  -u)  z (1  - v)  (1  - y) 
+ (1  - u)  (1  -z)  (1  - v)  y + (1  - u ) (1  - z)  (1  - v)  (1  - y), 
the  solution  of  this  system  of  equations  gives 
Prob.  w = cpq  + ac  (1  - q), 

whence 


Prob.  xy 
c 


= pq  + a (1  - q), 


the  value  required.  In  this  expression  the  arbitrary  constant  a 
is  the  probability  that  if  the  proposition  Z is  true  and  Y false,  X 
is  true.  In  other  words,  it  is  the  probability,  that  if  the  minor 
premiss  is  false,  the  conclusion  is  true. 

This  investigation  might  have  been  greatly  simplified  by  as- 
suming the  proposition  Zto  be  true,  and  then  seeking  the  proba- 
bility of  X.  The  data  would  have  been  simply 

Prob.  y = q,  Prob.  xy  =pq\ 

whence  we  should  have  found  Prob.  x = pq  + a (1  - q).  It  is 
evident  that  under  the  circumstances  this  mode  of  procedure 
would  have  been  allowable,  but  I have  preferred  to  deduce  the 
solution  by  the  direct  and  unconditioned  application  of  the 
method.  The  result  is  one  which  ordinary  reasoning  verifies, 
and  which  it  does  not  indeed  require  a calculus  to  obtain.  Ge- 
neral methods  are  apt  to  appear  most  cumbrous  when  applied  to 
cases  in  which  their  aid  is  the  least  required. 

Let  it  be  observed,  that  the  above  method  is  equally  appli- 
cable to  the  categorical  syllogism,  and  not  to  the  syllogism  only, 


286 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 


but  to  every  form  of  deductive  ratiocination.  Given  the  proba- 
bilities separately  attaching  to  the  premises  of  any  train  of  ar- 
gument ; it  is  always  possible  by  the  above  method  to  determine 
the  consequent  probability  of  the  truth  of  a conclusion  legitimately 
drawn  from  such  premises.  It  is  not  needful  to  remind  th§ 
reader,  that  the  truth  and  the  correctness  of  a conclusion  are  dif- 
ferent things. 

8.  One  remarkable  circumstance  which  presents  itself  in  such 
applications  deserves  to  be  specially  noticed.  It  is,  that  propo- 
sitions which,  when  true,  are  equivalent,  are  not  necessarily 
equivalent  when  regarded  only  as  probable.  This  principle  will 
be  illustrated  in  the  following  example. 

Ex.  6. — Given  the  probability  p of  the  disjunctive  proposition 
“ Either  the  proposition  Yis  true,  or  both  the  propositions  X and 
Y are  false,”  required  the  probability  of  the  conditional  propo- 
sition, “ If  the  proposition  X is  true,  Y is  true.” 

Let  x and  y be  appropriated  to  the  propositions  X and  Y 
respectively.  Then  we  have 

Prob.  y + (1  - x ) (1  - y)  = p, 

from  which  it  is  required  to  find  the  value  of  . 

^ Prob.® 


Assume  y + (1  - x)  (1  - y)  = t. 

Eliminating  y we  get 

(1  - x)  (1  - t)  = 0. 


0) 


whence 


0 


and  proceeding  in  the  usual  way, 

Prob.  x = 1 - p + cp.  (2) 

Where  c is  the  probability  that  if  either  Y is  true,  or  X and  Y 
false,  X is  true. 

Next  to  find  Prob.  xy.  Assume 
xy  = to. 

Eliminating  y from  (1)  and  (3)  we  get 

z (1  - t)  = 0 ; 


(3) 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS.  287 

whence,  proceeding  as  above, 

Prob.  z = cp, 

c having  the  same  interpretation  as  before.  Hence 

Prob.  xy  cp 
Prob.  x 1 - p + cp  ’ 

for  the  probability  of  the  truth  of  the  conditional  proposition 
giv  n. 

Now  in  the  science  of  pure  Logic,  which,  as  such,  is  conver- 
sant only  with  truth  and  with  falsehood,  the  above  disjunctive 
and  conditional  propositions  are  equivalent.  They  are  true  and 
they  are  false  together.  It  is  seen,  however,  from  the  above  in- 
vestigation, that  when  the  disjunctive  proposition  has  a proba- 
bility p,  the  conditional  proposition  has  a different  and  partly  in- 
cp 

definite  probability  - — Nevertheless  these  expressions 

are  such,  that  when  either  of  them  becomes  1 or  0,  the  other  as- 
sumes the  same  value.  The  results  are,  therefore,  perfectly  con- 
sistent, and  the  logical  transformation  serves  to  verify  the  formula 
deduced  from  the  theory  of  probabilities. 

The  reader  will  easily  prove  by  a similar  analysis,  that  if  the 
probability  of  the  conditional  proposition  were  given  as  p,  that 
of  the  disjunctive  proposition  would  be  1 - c + cp,  where  c is  the 
arbitrary  probability  of  the  truth  of  the  proposition  X. 

9.  Ex.  7. — Required  to  determine  the  probability  of  an  event 
x,  having  given  either  the  first,  or  the  first  and  second,  or  the 
first,  second,  and  third  of  the  following  data,  viz. : 

1st.  The  probability  that  the  event  x occurs,  or  that  it  alone 
of  the  three  events  x,  y,  z,  fails,  is  p. 

2nd.  The  probability  that  the  event  y occurs,  or  that  it  alone 
of  the  three  events  x,  y,  z,  fails,  is  q. 

3rd.  The  probability  that  the  event  z occurs,  or  that  it  alone 
of  the  three  events  x , y,  z , fails,  is  r. 

SOLUTION  OF  THE  FIRST  CASE. 

Here  we  suppose  that  only  the  first  of  the  above  data  is 


288 


ELEMENTARY  ILLUSTRATIONS. 


[chap.  XVIII. 


We  have  then, 


to  find  Prob.  x. 
Let 


Prob.  [x  + (1  - x)  yz } = p , 
x + (1  - x)  yz  = s, 


then  eliminating  yz  as  a single  symbol,  we  get, 

x (1  - s)  = 0. 


Hence 


whence,  proceeding  according  to  the  rule,  we  have 


Prob.  x = cp, 


0) 


where  c is  the  probability  that  if  x occurs,  or  alone  fails,  the 
former  of  the  two  alternatives  is  the  one  that  will  happen.  The 
limits  of  the  solution  are  evidently  0 and  p. 

This  solution  appears  to  give  us  no  information  beyond  what 
unassisted  good  sense  would  have  conveyed.  It  is,  however,  all 
that  the  single  datum  here  assumed  really  warrants  us  in  infer- 
ring. We  shall  in  the  next  solution  see  how  an  addition  to  our 
data  restricts  within  narrower  limits  the  final  solution. 


from  the  first  of  which  we  have,  by  (VIII.  7), 

[x  + (1  - x)yz)  (l-s)  + s{l-o:-(l- x)yz)  = 0, 


provided  that  for  simplicity  we  write  x for  1 - x,  y for  1 - y,  and 
so  on.  Now,  writing  for  1 - yz  its  value  in  constituents,  we 
have 

( x + xyz ) s + sx  (yz  + yz  + yz)  = 0, 
an  equation  consisting  solely  of  positive  terms. 


SOLUTION  OF  THE  SECOND  CASE. 

Here  we  assume  as  our  data  the  equations 


Prob.  [x  + (1  - x)  yz)  = p , 
Prob.  [y  + (1  -y)  xz } = q. 


Let  us  write 


x + (1  - x)  yz  = s, 
y + (\-y)xz  = f, 


or 


(x  + xyz)  s + sx  (1  - yz)  = 0 ; 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS.  289 

In  like  manner  we  have  from  the  second  equation, 

(y  + yxz)  t + ty  (xl  + xz  + xz)  = 0 ; 

and  from  the  sum  of  these  two  equations  we  are  to  eliminate  y 
and  z. 

If  in  that  sum  we  make  y = 1,  z = 1,  we  get  the  result  1 + 1. 
If  in  the  same  sum  we  make  y = 1,  z = 0,  we  get  the  result 

XS  + SX  + t. 

If  in  the  same  sum  we  make  y = 0,  z = 1,  we  get 
xl  + sx  + xt  + tx. 

And  if,  lastly,  in  the  same  sum  we  make  y = 0,  z = 0,  we  find 

xl  + sx  + tx  + tx,  or  xl  + sx  + t. 

These  four  expressions  are  to  be  multiplied  together.  New 
the  first  and  third  may  be  multiplied  in  the  following  manner : 

(1  + t ) (x~s  + sx  + xt  + tx) 

= xs  + xt  + (s  + t)  (sx  + tx)  by  (IX.  Prop,  ii.) 

= xs  + xt  + Ixt  + sxt.  (2) 

Again,  the  second  and  fourth  give  by  (IX.  Prop,  i.) 

(xl  + sx  + t)  (xl  + sx  + t) 

- x 1 + sx.  (3) 

Lastly,  (2)  and  (3)  multiplied  together  give 
(xl  + sx)  (xl  + sxt  + xt  + tx 1) 

= xl  + sx  (sxt  + xt  + tx 1) 

- xl  + sxt. 

Whence  the  final  equation  is 

(1  - s)  x + s (1  - t)  (1  -x)  = 0, 
which,  solved  with  reference  to  x,  gives 

« (1  - t) 

X~s(l-t)-(l-s) 

= ^ st  + s (1  _ 0 + 0 (1  - s)  t + 0 (1  - s)  (1  - t), 


290  ELEMENTARY  ILLUSTRATIONS.  [ciIAP.  XVIII. 

and,  proceeding  with  this  according  to  the  rule,  we  have,  finally, 

Prob.  x = p ( 1 - q)  + cpq.  (4) 

where  c is  the  probability  that  if  the  event  st  happen,  x will 
happen.  Now  if  we  form  the  developed  expression  of  st  by  mul- 
tiplying the  expressions  for  s and  t together,  we  find — 

c = Prob.  that  if  x and  y happen  together,  or  x and  z happen 
together,  and  y fail,  or  y and  z happen  together,  and  x fail,  the 
event  x will  happen. 

The  limits  of  Prob.  x are  evidently  p (1  - q)  and  p. 

This  solution  is  more  definite  than  the  former  one,  inasmuch 
as  it  contains  a term  unaffected  by  an  arbitrary  constant. 

SOLUTION  OF  THE  THIRD  CASE. 

Here  the  data  are — 

Prob.  [x  + (1  - x)  yz)  - p, 

Prob.  \y  + (1  - y)  xz)  = q, 

Prob.  [z  + (1  - z ) xy\  = r. 

Let  us,  as  before,  write  x for  1 - x , &c.,  and  assume 

x + xyz  = s, 
y + ijxz  = t, 
z + zxy  = u. 

On  reduction  by  (VIII.  8)  we  obtain  the  equation 
(x  + xyz')!  + sx  {yz  + yz  + yz) 

+ (y  + yxz)  t + ty  (zx  + xz  + xz) 

+ (z  + zxy)  u+  uz  (xy  + xy  + xy)  - 0.  (5) 

Now  instead  of  directly  eliminating  y and  z from  the  above 
equation,  let  us,  in  accordance  with  (IX.  Prop,  hi.),  assume  the 
result  of  that  elimination  to  be 

Ex  + E ( 1 - x)  = 0, 

then  E will  be  found  by  making  in  the  given  equation  x = 1, 
and  eliminating  y and  2 from  the  resulting  equation,  and  E will 
be  found  by  making  in  the  given  equation  x = 0,  and  eliminating 
y and  z from  the  result.  First,  then,  making  x = 1 , we  have 


CHAP.  XVIII.] 


ELEMENTARY  ILLUSTRATIONS. 


291 


s + (y  + yz)  t + tyz  + (z  + yz)  u + uyz  = 0, 

and  making  in  the  first  member  of  this  equation  successively 
y = 1,  z = 1,  y = l,  s = 0,  &c.,  and  multiplying  together  the 
results,  we  have  the  expression 

(7  + t + u ) ( 7 + t + u)  (7  + t + u)  (7  + t + «), 

which  is  equivalent  to 

(7  + t + u)  (7  + t + u). 

This  is  the  expression  for  E.  We  shall  retain  it  in  its  present 
form.  It  has  already  been  shown  by  example  (VTII.  3),  that 
the  actual  reduction  of  such  expressions  by  multiplication,  though 
convenient,  is  not  necessary. 

Again  in  (5),  making  x - 0,  we  have 


yzs  + s (yz  + yz  + yz)  + yt  + ty  + zu  + uz  - 0 ; 

from  which,  by  the  same  process  of  elimination,  we  find  for  E the 
expression 

(7+  t + u)  (s  + t + u)  (s  + t + u)  (s  + t + u). 

The  final  result  of  the  elimination  of  y and  z from  (5)  is  there- 
fore 

(7+i+w)(7+<+  w):r+(7+2+w)(s  + f+  w)(s+£+w)(s  + £+  u)(  1 -x)  = 0. 
Whence  we  have 

(7+?+w)  (s+£+w)  (s+£+w)  (s  + t+w) 
x — — — — — j 

(s+t  + u)  (s  + t + m)  (s  + t + u)  (5  + 1 + u)  -(s  + t+  w)(s  + £ + m) 


or,  developing  the  second  member, 

0 1 - 1 _ 

X = - stu  + — stu  + - stu  + stu 

.+  ^ 7 tu  + 07£w  + 07£w  + OlTu. 


(6) 


Hence,  passing  from  Logic  to  Algebra, 

stu  4 stu  stu  +7 tu  stu  + stu 

~P  q ~ r 

= stu  + stu  +7  tu  + stu  + stii. 


(7) 


292 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 


Prob.  x 


stu  + cstu 

stu  + stu  + !tu  + stu  + stu 


(8) 


• • • • s t 

To  simplify  this  system  of  equations,  change  — into  s,  — into  t, 

s t 

&c.,  and  after  the  change  let  X stand  for  stu  + s + t + 1 . We  then 
have 

s + cstu  , v 

— , (9) 


Prob.  x = 


with  the  relations 

stu  + s stu  + t stu  + u 


= stu + s+  t + u+  1=A.  (10) 

00 


p q r 

From  these  equations  we  get 

stu  + s - A p, 
stu  + s = X - t - u - 1, 

• Ayj  — A — u — t — 1, 
u + t = X (1  — p)  - 1. 

Similarly,  u + s = X (1  - q)  - 1, 

and  s + £ = A (1  - r)  -1. 

From  which  equations  we  find 

A(l+p-y-r)-l  \(l  + q-r-p)-l 

s = 2 ’ 1 = ~2  ’ 


u = 


Now,  by  (10), 


X(l  + r-  p-^)-l 


stu  - Xp  - s. 


(12) 


Substitute  in  this  equation  the  values  of  s,  t,  and  u above  deter- 
mined, and  we  have 

{(l+p-<7-r)X-l)((l-|-5,-p-r)A-l!{(l  + r-^-^)X-l| 

= 4((/>  + ? + r-l)A  + l),  (13) 

an  equation  which  determines  X.  The  values  of  s,  t,  and  u,  are 
then  given  by  (12),  and  their  substitution  in  (9)  completes  the 
solution  of  the  problem. 


CHAP.  XVIII.]  ELEMENTARY  ILLUSTRATIONS.  293 

10.  Now  a difficulty,  the  bringing  of  which  prominently  be- 
fore the  reader  has  been  one  object  of  this  investigation,  here 
arises.  How  shall  it  be  determined,  which  root  of  the  above 
equation  ought  to  taken  for  the  value  of  X.  To  this  difficulty 
some  reference  was  made  in  the  opening  of  the  present  chapter, 
and  it  was  intimated  that  its  fuller  consideration  was  reserved  for 
the  next  one ; from  which  the  following  results  are  taken. 

In  order  that  the  data  of  the  problem  may  be  derived  from 
a possible  experience,  the  quantities  p,  q,  and  r must  be  subject 
to  the  following  conditions  : 

\+p-q-rS>0, 

1 + ? ~P  ~ r > (14) 

\+r-p-q>0. 

Moreover,  the  value  of  X to  be  employed  in  the  general  solution 
must  satisfy  the  following  conditions  : 

X > t , X5. 1 , X > . (15) 

\ + p - q - r \ + q - p - r 1 + r - p - q 

Now  these  two  sets  of  conditions  suffice  for  the  limitation  of 
the  general  solution.  It  may  be  shown,  that  the  central  equation 
(13)  furnishes  but  one  value  of  X,  which  does  satisfy  these  con- 
ditions, and  that  value  of  X is  the  one  required. 

Let  1 + p - q - r be  the  least  of  the  three  coefficients  of  X 

given  above,  then  will  be  the  greatest  of  those  va- 

° i + p - q - r ° 

lues,  above  which  we  are  to  show  that  there  exists  but  one  value 

of  X.  Let  us  write  (13)  in  the  form 

{ ( 1 + p - q - r ) X - 1 ) { ( 1 + q - p - r)  X - 1 ) { ( 1 + r - p - q ) X - 1 ) 

-4  {(p  + q +r-l)  X + 1 } = 0 ; (16) 

and  represent  the  first  member  by  V. 

Assume  X = , then  V becomes 

l + p - q - r 

_i(P*i  + r-  !+1\  _4/ 2? \ 

\1  +p-q-r  j V+p-q-r) 

which  is  negative. 


294 


ELEMENTARY  ILLUSTRATIONS.  [CHAP.  XVIII. 

Let  X = oo,  then  V is  positive  and  infinite. 

Again, 

d-V 

— = (1  +p  - q - r)  (1  + q -p-  r)  {(1  + r -p  - q)\  - 1} 

+ similar  positive  terms, 

which  expression  is  positive  between  the  limits  X = 

c \ + p - q - r 

and  X = oo. 

If  then  we  construct  a curve  whose  abscissa  shall  be  measured 
by  X,  and  whose  ordinates  by  V,  that  curve  will,  between  the 
limits  specified,  pass  from  below  to  above  the  abscissa  X,  its  con- 
vexity always  being  downwards.  Hence  it  will  but  once  intersect 
the  abscissa X within  those  limits ; and  the  equation  (16)  will,  there- 
fore, have  but  one  root  thereto  corresponding. 

The  solution  is,  therefore,  expressed  by  (9),  X being  that 
root  of  (13)  which  satisfies  the  conditions  (15),  and  s,  t,  and  u 
being  given  by  (12).  The  interpretation  of  c may  be  deduced 
in  the  usual  way. 

It  appears  from  the  above,  that  the  problem  is,  in  all  cases, 
more  or  less  indeterminate. 


CHAP.  XIX.] 


OF  STATISTICAL  CONDITIONS. 


295 


CHAPTER  XIX. 

OF  STATISTICAL  CONDITIONS. 

1 . T)  Y the  term  statistical  conditions,  I mean  those  conditions 
-U  which  must  connect  the  numerical  data  of  a problem  in 
order  that  those  data  may  be  consistent  with  each  other,  and 
therefore  such  as  statistical  observations  might  actually  have 
furnished.  The  determination  of  such  conditions  constitutes  an 
important  problem,  the  solution  of  which,  to  an  extent  sufficient 
at  least  for  the  requirements  of  this  work,  I purpose  to  undertake 
in  the  present  chapter,  regarding  it  partly  as  an  independent  ob- 
ject of  speculation,  but  partly  also  as  a necessary  supplement  to 
the  theory  of  probabilities  already  in  some  degree  exemplified. 
The  nature  of  the  connexion  between  the  two  subjects  may  be 
stated  as  follows : 

2.  There  are  innumerable  instances,  and  one  of  the  kind 
presented  itself  in  the  last  chapter,  Ex.  7,  in  which  the  solution 
of  a question  in  the  theory  of  probabilities  is  finally  dependent 
upon  the  solution  of  an  algebraic  equation  of  an  elevated  degree. 
In  such  cases  the  selection  of  the  proper  root  must  be  determined 
by  certain  conditions,  partly  relating  to  the  numerical  values  as- 
signed in  the  data,  partly  to  the  due  limitation  of  the  element 
required.  The  discovery  of  such  conditions  may  sometimes  be 
effected  by  unaided  reasoning.  For  instance,  if  there  is  a proba- 
bility p of  the  occurrence  of  an  event  A,  and  a probability  q of 
the  concurrence  of  the  said  event  A,  and  another  event  B,  it  is 
evident  that  we  must  have 

P>B 

But  for  the  general  determination  of  such  relations,  a distinct 
method  is  required,  and  this  we  proceed  to  establish. 

As  derived  from  actual  experience,  the  probability  of  any 
event  is  the  result  of  a process  of  approximation.  It  is  the  limit 
of  the  ratio  of  the  number  of  cases  in  which  the  event  is  observed 
to  occur,  to  the  whole  number  of  equally  possible  cases  which 


296 


OF  STATISTICAL  CONDITIONS. 


[chap.  XIX. 

observation  records, — a limit  to  which  we  approach  the  more 
nearly  as  the  number  of  observations  is  increased.  Now  let  the 
symbol  n , prefixed  to  the  expression  of  any  class,  represent  the 
number  of  individuals  contained  in  that  class.  Thus,  x represent- 
ing men,  and  y white  beings,  let  us  assume 
nx  = number  of  men. 

nxy  = number  of  white  men. 

nx  (1  - y)  = number  of  men  who  are  not  white;  and  so  on. 

In  accordance  with  this  notation  w(l)  will  represent  the  number 
of  individuals  contained  in  the  universe  of  discourse,  and 

»0) 

will  represent  the  probability  that  any  individual  being,  selected 
out  of  that  universe  of  being  denoted  by  n (1),  is  a man.  If  ob- 
servation has  not  made  us  acquainted  with  the  total  values  of 
n (x)  and  ?i(l),  then  the  probability  in  question  is  the  limit  to 

Tl  ( X) 

which  approaches  as  the  number  of  individual  observations 
n (1)  11 

is  increased. 

In  like  manner  if,  as  will  generally  be  supposed  in  this  chap- 
ter, x represent  an  event  of  a particular  kind  observed,  n (x)  will 
represent  the  number  of  occurrences  of  that  event,  n(l)  the 
number  of  observed  events  (equally  probable)  of  all  kinds,  and 

, or  its  limit,  the  probability  of  the  occurrence  of  the 
event  x. 

Hence  it  is  clear  that  any  conclusions  which  may  be  deduced 
respecting  the  ratios  of  the  quantities  n ( x ),  n (_?/),  n (1),  &c.  may 
be  converted  into  conclusions  respecting  the  probabilities  of  the 
events  represented  by  x,  y,  &c.  Thus,  if  we  should  find  such  a 
relation  as  the  following,  viz., 

n(x)  + n(y)  < n(  1), 

expressing  that  the  number  of  times  in  which  the  event  x occurs 
and  the  number  of  times  in  which  the  event  y occurs,  are  toge- 
ther less  than  the  number  of  possible  occurrences  n (1),  we  might 
thence  deduce  the  relation, 

n (x)  n(y) 
n(lj  n(l) 

Prob.  x + Prob.  y < 1 . 


n (x) 
*0) 


or 


CHAP.  XIX.]  OF  STATISTICAL  CONDITIONS. 


297 


And  generally  any  such  statistical  relations  as  the  above  will  be 
converted  into  relations  connecting  the  probabilities  of  the  events 
concerned , by  changing  n(\)  into  1,  and  any  other  symbol  n (x) 
into  Prob.  x. 

3.  First,  then,  we  shall  investigate  a method  of  determining 
the  numerical  relations  of  classes  or  events,  and  more  particularly 
the  major  and  minor  limits  of  numerical  value.  Secondly,  we 
shall  apply  the  method  to  the  limitation  of  the  solutions  of  ques- 
tions in  the  theory  of  probabilities. 

It  is  evident  that  the  symbol  n is  distributive  in  its  operation. 
Thus  we  have 

n [xy  + (1  — x)  (1  - y) ) = nxy  + n (1  - x)  (1  - y ) 
nx  (1  - y)  - nx  - nxy, 

and  so  on.  The  number  of  things  contained  in  any  class  re- 
solvable into  distinct  groups  or  portions  is  equal  to  the  sum  of 
the  numbers  of  things  found  in  those  separate  portions.  It  is 
evident,  further,  that  any  expression  formed  of  the  logical  sym- 
bols x , y,  &c.  may  be  developed  or  expanded  in  any  way  consis- 
tent with  the  laws  of  the  symbols,  and  the  symbol  n applied  to 
each  term  of  the  result,  provided  that  any  constant  multiplier 
which  may  appear,  be  placed  outside  the  symbol  n;  without  affect- 
ing the  value  of  the  result.  The  expression  n (1),  should  it  ap- 
pear, will  of  course  represent  the  number  of  individuals  contained 
in  the  universe.  Thus, 

n (1-#)  (1  — 3/)  = w (1  - x - y + xy) 

= n(\)  - n O)  -n(y)  + n (xy). 

Again,  n [xy  + (1  - x)  (1  - y)}  = n (1  - x - y + 2 xy) 

= n (1)  - nx  - ny  + 2 nxy). 

In  the  last  member  the  term  2 nxy  indicates  twice  the  number  of 
individuals  contained  in  the  class  xy. 

4.  We  proceed  now  to  investigate  the  numerical  limits  of 
classes  whose  logical  expression  is  given.  In  this  inquiry  the 
following  principles  are  of  fundamental  importance  : 

1st.  If  all  the  members  of  a given  class  possess  a certain  pro- 
perty x,  the  total  number  of  individuals  contained  in  the  class  x 


OF  STATISTICAL  CONDITIONS. 


298 


[chap.  XIX. 


will  be  a superior  limit  of  the  number  of  individuals  contained  in 
the  given  class. 

2nd.  A minor  limit  of  the  number  of  individuals  in  any  class  y 
will  be  found  by  subtracting  a major  numerical  limit  of  the  con- 
trary class,  1 - y,  from  the  number  of  individuals  contained  in  the 
universe. 

To  exemplify  these  principles,  let  us  apply  them  to  the  fol- 
lowing problem : 

Problem. — Given,  n(l),  n (x),  and  n(y),  required  the  su- 
perior and  inferior  limits  of  nxy. 

Here  our  data  are  the  number  of  individuals  contained  in  the 
universe  of  discourse,  the  number  contained  in  the  class  x,  and 
the  number  in  the  class  y,  and  it  is  required  to  determine  the 
limits  of  the  number  contained  in  the  class  composed  of  the  indi- 
viduals that  are  found  at  once  in  the  class  x and  in  the  class  y. 

By  Principle  i.  this  number  cannot  exceed  the  number  con- 
tained in  the  class  x,  nor  can  it  exceed  the  number  contained  in 
the  class  y.  Its  major  limit  will  then  be  the  least  of  the  two  va- 
lues n (x)  and  (y). 

By  Principle  n.  a minor  limit  of  the  class  xy  will  be  given  by 
the  expression 

w(l)  -major  limit  of  {r(l  -y)  + y(  1 - x)  + (1  - x)  (1  - y )},  (1) 

since  x (1  - y)  + y (1  - r)  + (1  - x)  (1  - y)  is  the  complement  of 
the  class  xy,  i.  e.  what  it  wants  to  make  up  the  universe. 

Now  x{\  - y)  + (1  - x)  (1  - y)  = 1 - y.  We  have  there- 
fore for  (1), 

n (l)  - major  limit  of  { 1 -y  + y (1  - x)) 

= n (1)  - n (1  - y)  - major  limit  of  y (1  - x ).  (2) 

The  major  limit  of  y (1  - x)  is  the  least  of  the  two  values  n (y) 
and  n (1  - x).  Let  n ( y ) be  the  least,  then  (2)  becomes 

n(l)-n(l-y)-n(y) 

= w(l)  - w(l)  + n(y ) - n(y ) = 0. 

Secondly,  let  n (1  - x)  be  less  than  n (y),  then 
major  limit  of  ny  (1  - x~)  = n{\  - x) ; 
therefore  (2)  becomes 


CHAP.  XIX.] 


OF  STATISTICAL  CONDITIONS. 


299 


n (1)  - n(\  - y)  - n(\  - x) 

= n (1)  - n (1)  + n ( y ) - n (1)  + n ( x ) 

= nx  + ny  - n (1). 

The  minor  limit  of  nxy  is  therefore  either  0 or  n (x)  + n (y)  - w(l), 
according  as  n ( y ) is  less  or  greater  than  n (1  - x),  or,  which  is  an 
equivalent  condition,  according  as  n ( x ) is  greater  or  less  than 
n(\-y). 

Now  as  0 is  necessarily  a minor  limit  of  the  numerical  value 
of  any  class,  it  is  sufficient  to  take  account  of  the  second  of  the 
above  expressions  for  the  minor  limit  of  n (xy).  We  have,  there- 
fore, 

Major  limit  of  n (xy)  = least  of  values  n (x)  and  n (y). 

Minor  limit  of  n (xy)  - n (x)  + n(y)  - n(\).*  • 

Proposition  I. 

5.  To  express  the  major  and  minor  limits  of  a class  represented 
by  any  constituent  of  the  symbols  x,  y,  z,  frc.,  having  given  the  va- 
lues ofn(x),  n(y),  n(z),  SfC;,  and  «(1). 

Consider  first  the  constituent  xyz. 

It  is  evident  that  the  major  numerical  limit  will  be  the  least 
of  the  values  n(x),  n(y),  n(z). 

The  minor  numerical  limit  may  be  deduced  as  in  the  previous 
problem,  but  it  may  also  be  deduced  from  the  solution  of  that 
problem.  Thus : 

Minor  limit  of  n (xyz)  = n (xy)  + n(z)  - n(  1).  (1) 

Now  this  means  that  n (xyz)  is  at  least  as  great  as  the  expres- 
sion n(xy)  + n(z)  - w(l).  But  n(xy)  is  at  least  as  great  as 
n (x)  + n(y)  - n (1).  Therefore  n (xyz)  is  at  least  as  great  as 

n (x)  + n (y)  - n (1)  + n (z)  - n (1), 
or  n (a’)  + n (y)  + n (z)  - 2 n (1). 


* The  above  expression  for  the  minor  limit  of  nxy  is  applied  by  Professor 
De  Morgan,  by  whom  it  appears  to  have  been  first  given,  to  the  syllogistic  form  : 
Most  men  in  a certain  company  have  coats. 

Most  men  in  the  same  company  have  waistcoats. 

Therefore  some  in  the  company  have  coats  and  waistcoats. 


300 


OF  STATISTICAL  CONDITIONS. 


[chap.  XIX. 


Hence  we  have 

Minor  limit  of  n ( xyz ) = n (x)  + n (y)  + n (z)  - 2n  (1). 

By  extending  this  mode  of  reasoning  we  shall  arrive  at  the 
following  conclusions : 

1st.  The  major  numerical  limit  of  the  class  represented  by 
any  constituent  will  be  found  by  prefixing  n separately  to  each 
factor  of  the  constituent,  and  taking  the  least  of  the  resulting 
values. 

2nd.  The  minor  limit  will  be  found  by  adding  all  the  values 
above  mentioned  together,  and  subtracting  from  the  result  as 
many,  less  one,  times  the  value  of  ra(l). 

Thus  we  should  have 

Major  limit  of  nxy  (1  - z)  = least  of  the  values  nx,  ny , and  n{\  -z). 
Minor  limit  of  nxy  (1  - z)  = n (x)  + n (y)  + n (1  - z)  - 2n{\) 

= nx  + n(y)  - n ( z ) - n{  1). 

In  the  use  of  general  symbols  it  is  perhaps  better  to  regard  all 
the  values  n (x),  n (y),  n (1  - z),  as  major  limits  of  n [xy  (1  - z)}, 
since,  in  fact,  it  cannot  exceed  any  of  them.  I shall  in  the  fol- 
lowing investigations  adopt  this  mode  of  expression. 

Proposition  II. 

6.  To  determine  the  major  numerical  limit  of  a class  expressed 
by  a series  of  constituents  of  the  symbols  x,  y,  z,  Sfc.,  the  values  of 
n (x),  n (y),  n(z ),  §-c.,  and  n (1),  being  given. 

Evidently  one  mode  of  determining  such  a limit  would  be  to 
form  the  least  possible  sum  of  the  major  limits  of  the  several  con- 
stituents. Thus  a major  limit  of  the  expression 

n{xy+  (1  - x)  (1  - y)) 

would  be  found  by  adding  the  least  of  the  two  values  nx , ny , fur- 
nished by  the  first  constituent,  to  the  least  of  the  two  values 
n (1  - x),  n (1  - y ),  furnished  by  the  second  constituent.  If  we 
do  not  know  which  is  in  each  case  the  least  value,  we  must  form 
the  four  possible  sums,  and  reject  any  of  these  which  are  equal  to 
or  exceed  n (1).  Thus  in  the  above  example  we  should  have 


CHAP.  XIX.] 


OF  STATISTICAL  CONDITIONS. 


301 


nx  + n(l  - x)  = n (1). 
n(x ) + «(1  -y)  = «(1)  + n(x)  - n(y). 
n(y)  +n(l-y)  = n(\)  + n(y)  -n(x). 
n(y)  + «(1  - y)  = ra(l). 

Rejecting  the  first  and  last  of  the  above  values,  we  have 
n (1)  + n (x)  - n (y),  and  n (1)  + n {y)  - n ( x ), 
for  the  expressions  required,  one  of  which  will  (unless  nx  = ny) 
be  less  than  ra(l),  and  the  other  greater.  The  least  must  of 
course  be  taken. 

When  two  or  more  of  the  constituents  possess  a common  fac- 
tor, as  x,  that  factor  can  only,  as  is  obvious  from  Principle  I., 
furnish  a single  term  n (x)  in  the  final  expression  of  the  major 
limit.  Thus  if  n (x)  appear  as  a major  limit  in  two  or  more  con- 
stituents, we  must,  in  adding  those  limits  together,  replace 
nx  + nx  by  nx,  and  so  on.  Take,  for  example,  the  expression 

n [xy  + x (1  - y)z)  ■ The  major  limits  of  this  expression,  imme- 

diately furnished  by  addition,  would  be — 

1.  nx.  4.  ny  + nx. 

2.  nx  + n (1  - y).  5.  ny  + n (1  - y). 

3.  nx  + n (z).  6.  ny  + nz. 

Of  these  the  first  and  sixth  only  need  be  retained ; the  second, 
third,  and  fourth  being  greater  than  the  first ; and  the  fifth  being 
equal  to  n (1).  The  limits  are  therefore 

n ( x ) and  n (y)  + n ( z ), 

and  of  these  two  values  the  last,  supposing  it  to  be  less  than  n (1), 
must  be  taken. 

These  considerations  lead  us  to  the  following  Rule : 

Rule. — Take  one  factor  from  each  constituent,  and  prefx  to 
it  the  symbol  n,  add  the  several  terms  or  results  thus  formed  toge- 
ther, rejecting  all  repetitions  of  the  same  term  ; the  sum  thus  ob- 
tained will  be  a major  limit  of  the  expression,  and  the  least  of  all 
such  sums  ivill  be  the  major  limit  to  be  employed. 

Thus  the  major  limits  of  the  expression 

xyz  + #(1  -y)  (l  - z)  + (1  - *)  (1  - y)  (1  - z) 
would  be 


302 


OF  STATISTICAL  CONDITIONS.  [CHAP.  XIX. 

n (#)  + n (1  - y),  and  n(x)  + n{  1 - z), 
or  n (x)  + n (1)  - n (y),  and  n (x)  + n (1)  - n (z). 

If  we  began  with  n (y),  selected  from  the  first  term,  and  took 
n (x)  from  the  second,  we  should  have  to  take  n (1  -y)  from  the 
third  term,  and  this  would  give 

n (y)  + n (x)  + n (1  - y),  or  n (1)  + n (#). 

But  as  this  result  exceeds  n (1),  which  is  an  obvious  major  limit 
to  every  class,  it  need  not  be  taken  into  account. 

Proposition  III. 

7.  To  find  the  minor  numerical  limit  of  any  class  expressed  by 
constituents  of  the  symbols  x,  y,  z,  having  given  n(x),  n(y),  n(z)  . . 
n(l). 

This  object  may  be  effected  by  the  application  of  the  pre- 
ceding Proposition,  combined  with  Principle  ii.,  but  it  is  better 
effected  by  the  following  method  : 

Let  any  two  constituents,  which  differ  from  one  another  only 
by  a single  factor,  be  added,  so  as  to  form  a single  class  term 
as  x (1  -y)  + xy  form  x,  and  this  species  of  aggregation  having 
been  carried  on  as  far  as  possible,  i.  e.,  there  having  been  selected 
out  of  the  given  series  of  constituents  as  many  sums  of  this  kind 
as  can  be  formed,  each  such  sum  comprising  as  many  constituents 
as  can  be  collected  into  a single  term,  without  regarding  whether 
any  of  the  said  constituents  enter  into  the  composition  of  other 
terms,  let  these  ultimate  aggregates,  together  with  those  con- 
stituents which  do  not  admit  of  being  thus  added  together,  be 
written  down  as  distinct  terms.  Then  the  several  minor  limits 
of  those  terms,  deduced  by  Prop.  I.,  will  be  the  minor  limits  of 
the  expression  given,  and  one  only  of  those  minor  limits  will  at 
the  same  time  be  positive. 

Thus  from  the  expression  xy  + (1  - x)y  + (1  - x)  (1  -y)  we 
can  form  the  aggregates  y and  1 - x,  by  respectively  adding  the 
first  and  second  terms  together,  and  the  second  and  third. 
Hence  n (y)  and  n(l  - x)  will  be  the  minor  limits  of  the  expres- 
sion given.  Again,  if  the  expression  given  were 


303 


CRAP.  XIX.]  OF  STATISTICAL  CONDITIONS. 

xyz  + x (1  — y)  z + (1  — x)  yz  + (1  - x)  (1  - y)  z 

+ xy  (1  - z)  + (1  - x)  (1  - y ) (1  - z), 

we  should  obtain  by  addition  of  the  first  four  terms  the  single 
term  z,  by  addition  of  the  first  and  fifth  term  the  single  term  xy , 
and  by  addition  of  the  fourth  and  sixth  terms  the  single  term 
(1  - Xs)  (1  - y) ; and  there  is  no  other  way  in  which  constituents 
can  be  collected  into  single  terms,  nor  are  there  are  any  consti- 
tuents left  which  have  not  been  thus  taken  account  of.  The 
three  resulting  terms  give,  as  the  minor  limits  of  the  given  ex- 
pression, the  values 

n(z),  n (x)  + n(y)  - «(1), 

and  n (1  - x)  + n (1  - y)  - n (1),  or  n (1)  - n(x)  - n (y). 

8.  The  proof  of  the  above  rule  consists  in  the  proper  appli- 
cation of  the  following  principles  : — 1st.  The  minor  limit  of  any 
collection  of  constituents  which  admit  of  being  added  into  a sin- 
gle term,  will  obviously  be  the  minor  limit  of  that  single  term. 
This  explains  the  first  part  of  the  rule.  2nd.  The  minor  limit 
of  the  sum  of  any  two  terms  which  either  are  distinct  constituents, 
or  consist  of  distinct  constituents,  but  do  not  admit  of  being 
added  together,  will  be  the  sum  of  their  respective  minor  limits, 
if  those  minor  limits  are  both  positive ; but  if  one  be  positive,  and 
the  other  negative,  it  will  be  equal  to  the  positive  minor  limit 
alone.  For  if  the  negative  one  were  added,  the  value  of  the  limit 
would  be  diminished,  i.  e.  it  would  be  less  for  the  sum  of  two 
terms  than  for  a single  term.  Now  whenever  two  constituents 
dilfer  in  more  than  one  factor,  so  as  not  to  admit  of  being  added 
together,  the  minor  limits  of  the  two  cannot  be  both  positive. 
Thus  let  the  terms  be  xyz  and  ( 1 - x)  ( 1 - y)z,  which  differ  in 
two  factors,  the  minor  limit  of  the  first  is  n(x  + y + z-  2),  that 
of  the  second  n (1  - x + 1 - y + z - 2),  or, 

1st.  n {x  + y - 1 - (1,-z)).  2nd.  n {1  - x - y - (1  - z)} . 

If  n(x  + y - 1)  is  positive,  n{\  - x - y)  is  negative,  and  the  se- 
cond must  be  negative.  If  n (x  + y - 1)  is  negative,  the  first  is 
negative;  and  similarly  for  cases  in  which  a larger  number  of 
factors  are  involved.  It  may  in  this  manner  be  shown  that,  ac- 
cording to  the  mode  in  which  the  aggregate  terms  are  formed  in 


304 


OF  STATISTICAL  CONDITIONS.  [CHAP.  XIX. 

the  application  of  the  rule,  no  two  minor  limits  of  distinct  terms 
can  be  added  together,  for  either  those  terms  will  involve  some 
common  constituent,  in  which  case  it  is  clear  that  we  cannot  add 
their  minor  limits  together, — or  the  minor  limits  of  the  two  will 
not  be  both  positive,  in  which  case  the  addition  would  be  useless. 


Proposition  IY. 


9.  Given  the  respective  numbers  of  individuals  comprised  in 
any  classes,  s,  t,  Sfc.  logically  defined,  to  deduce  a system  of  nume- 
rical limits  of  any  other  class  w,  also  logically  defined. 


As  this  is  the  most  general  problem  which  it  is  meant  to  dis- 
cuss in  the  present  chapter,  the  previous  inquiries  being  merely 
introductory  to  it,  and  the  succeeding  ones  occupied  with  its  ap- 
plication, it  is  desirable  to  state  clearly  its  nature  and  design. 

When  the  classes  s,  t . . w are  said  to  be  logically  defined,  it 
is  meant  that  they  are  classes  so  defined  as  to  enable  us  to  write 
down  their  symbolical  expressions,  whether  the  classes  in  ques- 
tion be  simple  or  compound.  By  the  general  method  of  this 
treatise,  the  symbol  w can  then  be  determined  directly  as  a deve- 
loped function  of  the  symbols  s,  t,  &c.  in  the  form 


0 1 

iv  = A + OB  + - C + - D, 


(1) 


wherein  A,  B,C,  and  D are  formed  of  the  constituents  of  s,  t,  &c. 
How  from  such  an  expression  the  numerical  limits  of  w may  in 
the  most  general  manner  be  determined,  will  be  considered  here- 
after. At  present  we  merely  purpose  to  show  how  far  this  object 
can  be  accomplished  on  the  principles  developed  in  the  previous 
propositions;  such  an  inquiry  being  sufficient  for  the  purposes  of 
this  work.  For  simplicity,  I shall  found  my  argument  upon  the 
particular  development, 

w = st  + 0«  (1  - t)  + - (1  - s)  t + - (1  - s)  (l  - t),  (2) 


in  which  all  the  varieties  of  coefficients  present  themselves. 

Of  the  constituent  (l  - s)  (1  - t),  which  has  for  its  coeffi- 
cient jj,  it  is  implied  that  some,  none,  or  all  of  the  class  denoted 


305 


CHAP.  XIX.]  OF  STATISTICAL  CONDITIONS. 

by  that  constituent  are  found  in  w.  It  is  evident  that  n (w)  will 
have  its  highest  numerical  value  when  all  the  members  of  the 
class  denoted  by  (1  -s)  (1  -t)  are  found  in  w.  Moreover,  as 
none  of  the  individuals  contained  in  the  classes  denoted  by 
s (1  - 1)  and  (1  - s)  t are  found  in  to,  the  superior  numerical  limits 
of  tv  will  be  identical  with  those  of  the  class  st  + (1  - s)  (1  - t). 
They  are,  therefore, 

ns  + n (1  - t ) and  nt  + n (1  - s ). 

In  like  manner  a system  of  superior  numerical  limits  of  the 
development  A + OB  + ^ C + - D,  may  be  found  from  those  of 
A + C by  Prop.  2. 

Again,  any  minor  numerical  limit  of  tv  will,  by  Principle  n., 
be  given  by  the  expression 

n (1)  - major  limit  of  n (1  - to), 

but  the  development  ofw;  being  given  by  (1),  that  of  1 - to  will 
obviously  be 

1 -w  = 0A  + B + + ^Z>. 

This  may  be  directly  proved  by  the  method  of  Prop.  2,  Chap.  x. 
Hence 

Minor  limit  of  n (w)  = n (1)  - major  limit  (H  + C) 

= minor  limit  of  (A  + D ), 

by  Principle  ii.,  since  the  classes  A -f  I)  and  B + C are  supple- 
mentary. Thus  the  minor  limit  of  the  second  member  of  (2) 
would  be  n (f),  and,  generalizing  this  mode  of  reasoning,  we  have 
the  following  result : 

A system  of  minor  limits  of  the  development 
A + OB + ^C+ 

will  be  given  by  the  minor  limits  of  A + D. 

This  result  may  also  be  directly  inferred.  For  of  minor  nu- 
merical limits  we  are  bound  to  seek  the  greatest.  Now  we  ob- 
tain in  general  a higher  minor  limit  by  connecting  the  class  D 


306 


OF  STATISTICAL  CONDITIONS* 


[CHAP.  XIX. 


with  A in  the  expression  of  w.  a combination  which,  as  shown  in 
various  examples  of  the  Logic  we  are  permitted  to  make,  than 
we  otherwise  should  obtain. 

Finally,  as  the  concluding  term  of  the  development  of  w in- 
dicates the  equation  D = 0,  it  is  evident  that  re  (D)  = 0.  Hence 
we  have 

Minor  limit  of  re  (IJ)  < 0, 

and  this  equation,  treated  by  Prop.  3,  gives  the  requisite  condi- 
tions among  the  numerical  elements  re(s),  n(t ),  &c.,  in  order  that 
the  problem  may  be  real,  and  may  embody  in  its  data  the  re- 
sults of  a possible  experience. 

Thus  from  the  term  ^ ( 1 - s)  t in  the  second  member  of  (2) 
we  should  deduce 

re  ( 1 - s)  + re  (t)  - re  (1)  < 0, 

•\  n ( l ) < re(s). 

These  conclusions  may  be  embodied  in  the  following  rule  : 

10.  Pule. — Determine  the  expression  of  the  class  w as  a deve- 
loped logical  function  of  the  symbols  s,  t,  Sfc.  in  the  form 

w = A + 0B  + yC+}-D. 

0 0 

Then  will 

Maj.  lim,  w = Maj.  lim,  A + C. 

Min,  lim.  w = Min.  lim.  A + D. 

The  necessary  numerical  conditions  among  the  data  being  given  by 
the  inequality 

Min.  lim.  J5  < w (1). 

To  apply  the  above  method  to  the  limitation  of  the  solutions 
of  questions  in  probabilities,  it  is  only  necessary  to  replace  in 
each  of  the  formula,  n (x)  by  Prob.  x,  n (y)  by  Prob.  y,  &c.,  and, 
finally,  re  (1)  by  1.  The  application  being,  however,  of  great  im- 
portance, it  may  be  desirable  to  exhibit  in  the  form  of  a rule 
the  chief  results  of  transformation. 

11.  Given  the  probabilities  of  any  events  s,  t,  &c.,  whereof 
another  event  is  a developed  logical  function,  in  the  form 

w = A + 0B  + ? C + l D, 

0 0 


OF  STATISTICAL  CONDITIONS. 


CHAP.  XIX.] 


307 


required  the  systems  of  superior  and  inferior  limits  of  Prob.  w, 
and  the  conditions  among  the  data. 

Solution. — The  superior  limits  of  Prob.  ( A + C ),  and  the 
inferior  limits  of  Prob.  (A  + D)  will  form  two  such  systems  as  are 
sought.  The  conditions  among  the  constants  in  the  data  will  be 
given  by  the  inequality, 

Inf.  lim.  Prob.  D < 0. 


In  the  application  of  these  principles  we  have  always 
Inf.  lim.  Prob.  xx  x2 . . x„  = Prob.  xx  + Prob.  xt..+  Prob.  xn  - (n  - 1 ). 

Moreover,  the  inferior  limits  can  only  be  determined  from  single 
terms,  either  given  or  formed  by  aggregation.  Superior  limits 
are  included  in  the  form  S Prob.  x,  Prob.  x applying  only  to 
symbols  which  are  different,  and  are  taken  from  different  terms  in 
the  expression  whose  superior  limit  is  sought.  Thus  the  supe- 
rior limits  of  Prob.  xyz  + x (1  - y)  (1  - z)  are 

Prob. a;,  Prob.?/  + Prob.  (1  - z),  and  Prob.  z + Prob.  (1  -y). 

Let  it  be  observed,  that  if  in  the  last  case  we  had  taken  Prob.  z 
from  the  first  term,  and  Prob.  (1  - z)  from  the  second, — a con- 
nexion not  forbidden, — we  should  have  had  as  their  sum  1,  which 
as  a result  would  be  useless  because  d priori  necessary.  It  is 
obvious  that  we  may  reject  any  limits  which  do  not  fall  between 
0 and  1. 

Let  us  apply  this  method  to  Ex.  7,  Case  hi.  of  the  last 
chapter. 

The  final  logical  solution  is 

0 , 1 - 1 . 

X - - stu  + - stu  + - stu  + stu 
0 0 0 

+ \-ltu+  0 iftu+  b~stu  + 0!tu, 

0 

the  data  being 

Prob.  s = p,  Prob.  t = q,  Prob.  u = r. 

We  shall  seek  both  the  numerical  limits  of  x,  and  the  condi- 
tions connecting  p,  q,  and  r. 


308  OF  STATISTICAL  CONDITIONS.  [CHAP.  XIX. 

The  superior  limits  of  x are,  according  to  the  rule,  given  by 
those  of  stu  + stu.  They  are,  therefore, 

p,  q + 1 — r,  r + 1 - q. 

The  inferior  limits  of  x arc  given  by  those  of 
stu  + stu  + stu  + Itu. 

We  may  collect  the  first  and  third  of  these  constituents  in  the 
single  term  st,  and  the  second  and  third  in  the  single  term  su. 
The  inferior  limits  of  x must  then  be  deduced  separately  from 
the  terms  s ( l - £),  5 (1  - u),  (1  - s)  tu , which  give 

p+l-q-\,  p+l-r-l,  l-p  + q + r-2, 
or  p - q,  p - r,  and  q + r - p - 1 . 

Finally,  the  conditions  among  the  constants  p,  q,  and  r,  are 
given  by  the  terms 

stu,  stu,  stu , 

from  which,  by  the  rule,  we  deduce 

p+\-q  + r-2<0,  p + q + \-r-2<0,  \ - p+ q + r -2<0. 
or  \ + q - p - r>0,  1 + r - p - q>0,  1 + p - q - r>0. 

These  are  the  limiting  conditions  employed  in  the  analysis  of 
the  final  solution.  The  conditions  by  which  in  that  solution  A is 
limited,  were  determined,  however,  simply  from  the  conditions 
that  the  quantities  s,  t,  and  u should  be  positive.  Narrower 
limits  of  that  quantity  might,  in  all  probability,  have  been  de- 
duced from  the  above  investigation. 

12.  The  following  application  is  taken  from  an  important  pro- 
blem, the  solution  of  which  will  be  given  in  the  next  chapter. 
There  are  given, 

Prob.  x = c{,  Prob. y - c2,  Prob.  s = cxpx,  Prob . t = c2p», 
together  with  the  logical  equation 

z = stocy  + stxy  + stxy  + 07£ 

1 f stxy  + stxy  + stxy  + stxy  + stxy 
® {_  + s txy  + ~stxy  + Itxy  + stxy ; 


CHAP.  XIX.] 


OF  STATISTICAL  CONDITIONS. 


309 


and  it  is  required  to  determine  the  conditions  among  the  constants 
Ci,  c2,  Pn  Pz,  and  the  major  and  minor  limits  of  z. 

First  let  us  seek  the  conditions  among  the  constants.  Con- 


fining our  attention  to  the  terms  whose  coefficients 


1 

are  - , we 


readily  form,  by  the  aggregation  of  constituents,  the  following 
terms,  viz. : 

5(1-*),  t(l-y),  sy(\-t),  ta(l-s); 


nor  can  we  form  any  other  terms  which  are  not  included  under 
these.  Hence  the  conditions  among  the  constants  are, 


n (s)  + n (1  - a)  - n (1)  < 0, 

»(0  +»(1  -y)  ~ »(1)  < 0, 

n (5)  + n ( y ) + n(\  - t)  - 2n  (1)  < 0, 
n(t)  + n (a)  + n (1  - s)  - 2 n (1)  < 0. 

Now  replace  n (a)  by  cls  n (; y ) by  c2,  n ( s ) by  c,/*!,  n ( t ) by 
c2p2,  and  w(l)  by  1,  and  we  have,  after  slight  reductions, 

C1P1  < Ci , c2y>2  < £25 

C1P1  < 1 - c2  (1  -p2),  c2p2  < 1 - Cj  (1  -pv). 

Such  are,  then,  the  requisite  conditions  among  the  constants. 

Again,  the  major  limits  of  z are  identical  Avith  those  of  the 
expression 

stxy  + s ( \ - t)  x ( \ - y)  + - s)  t ( \ - x)  y\ 

which,  if  Ave  bear  in  mind  the  conditions 

n (s)  < n (a),  n (t)  < n (y), 


above  determined,  will  be  found  to  be 


n(s)  + n ( t ),  or,  cL + c.,p2, 

n (s)  + n (1  - a),  or,  1 - c2  (1  - px) 
n(t)  + h (1  - y),  or,  1 - c2  (1  - p2). 

Lastly,  to  ascertain  the  minor  limits  of  2,  Ave  readily  form 
from  the  constituents,  Avhose  coefficients  are  1 or  the  single 
terms  s and  t,  nor  can  any  other  terms  not  included  under  these  be 


OF  STATISTICAL  CONDITIONS. 


310 


[chap.  XIX. 


formed  by  selection  or  aggregation.  Hence,  for  the  minor  limits 
of  z we  have  the  values  cxpx  and  c2p2 . 

13.  It  is  to  be  observed,  that  the  method  developed  above 
does  not  always  assign  the  narrowest  limits  which  it  is  possible 
to  determine.  But  it  in  all  cases,  I believe,  sufficiently  limits  the 
solutions  of  questions  in  the  theory  of  probabilities. 

The  problem  of  the  determination  of  the  narrowest  limits  of 
numerical  extension  of  a class  is,  however,  always  reducible  to  a 
purely  algebraical  form,*  Thus,  resuming  the  equations 

iv  = A + OB  + ^ C + ^ D, 

0 0 

let  the  highest  inferior  numerical  limit  of  w be  represented  by 
the  formula  an  (&•)  + bn  (t)  . . + dn  (1),  wherein  a,  b,  c,  ..  d are 
numerical  constants  to  be  determined,  and  s,  f,  &c.,  the  logical 
symbols  of  which  A,  B,  C,  D are  constituents.  Then 

an  (s)  + bn  (t)  . . + dn  (1)  = minor  limit  of  A subject 

to  the  condition  D = 0. 

Hence  if  we  develop  the  function 

as  + bt . . + <f, 

reject  from  the  result  all  constituents  which  are  found  in  D , the 
coefficients  of  those  constituents  which  remain,  and  are  found 
also  in  A,  ought  not  individually  to  exceed  unity  in  value,  and 
the  coefficients  of  those  constituents  which  remain,  and  which 
are  not  found  in  A,  should  individually  not  exceed  0 in  value. 
Hence  we  shall  have  a series  of  inequalities  of  the  form  f<  1 , 
and  another  series  of  the  form  g < 0;,  /’and  g being  linear  func- 
tions of  a,  b , c,  &c.  Then  those  values  of  a,  b . . d,  which,  while 
satisfying  the  above  conditions,  give  to  the  function 

an  (5)  + bn  ( t ) . . + dn  (1), 

its  highest  value  must  be  determined,  and  the  highest  value  in 


* The  author  regrets  the  loss  of  a manuscript,  written  about  four  years  ago, 
in  which  this  method,  he  believes,  was  developed  at  considerable  length.  His 
recollection  of  the  contents  is  almost  entirely  confined  to  the  impression  that  the 
principle  of  the  method  was  the  same  as  above  described,  and  that  its  suffici- 
ency was  proved.  The  prior  methods  of  this  chapter  are,  it  is  almost  needless 
to  say,  easier,  though  certainly  less  general. 


CHAP.  XIX.] 


OF  STATISTICAL  CONDITIONS. 


311 


question  will  be  the  highest  minor  limit  of  id.  To  the  above  we 
may  add  the  relations  similarly  formed  for  the  determination  of 
the  relations  among  the  given  constants  n (s),  n(t)  . ,n  (1). 

14.  The  following  somewhat  complicated  example  will  show 
how  the  limitation  of  a solution  is  effected,  when  the  problem 
involves  an  arbitrary  element,  constituting  it  the  representative 
of  a system  of  problems  agreeing  in  their  data,  but  unlimited  in 
their  quaesita. 

Problem. — Of  n events  xx  x2 . . xn,  the  following  particulars 
are  known : 

1st.  The  probability  that  either  the  event  xx  will  occur,  or 
all  the  events  fail,  is  px . 

2nd.  The  probability  that  either  the  event  x2  will  occur,  or 
all  the  events  fail,  is  p2 . And  so  on  for  the  others. 

It  is  required  to  find  the  probability  of  any  single  event,  or 
combination  of  events,  represented  by  the  general  functional  form 
$(xx  . . xn),  or  0. 

Adopting  a previous  notation,  the  data  of  the  problem  are 
Prob.  (xx  + ij  . . xn)  = Pi  . . Prob.  (xn  + ix  . . xn)  = pn . 

And  Prob.  0 (xx  . . xn)  is  required. 

Assume  generally 

xr  + xl  . . xn  = sr,  (1) 

<p  = w.  (2) 

We  hence  obtain  the  collective  logical  equation  of  the  problem 

2 {(xr  + x1  . . xn)  Jr  + sr  (x,.  - xx  . . •?„)}  + <j>w  + vity  = 0.  (3) 

From  this  equation  we  must  eliminate  the  symbols  xx, . . xn,  and 
determine  w as  a developed  logical  function  of  sx . . sn . 

Let  us  represent  the  result  of  the  aforesaid  elimination  in  the 
form 

Ew  + A'(l  - to)  = 0 ; 

then  will  E be  the  result  of  the  elimination  of  the  same  symbols 
from  the  equation 

S {(ay  + xx , . x„)  Jr  + sr  (xr  -xl..xn))+  1-0  = 0.  (4) 

Now  E will  be  the  product  of  the  coefficients  of  all  the  con- 
stituents (considered  with  reference  to  the  symbols  x1}  x.2  . . xn) 


312  OF  STATISTICAL  CONDITIONS.  [CHAP.  XIX. 

which  nre  found  in  the  development  of  the  first  member  of  the 
above  equation.  Moreover,  0,  and  therefore  1-0,  will  consist 
of  a series  of  such  constituents,  having  unity  for  their  respective 
coefficients.  In  determining  the  forms  of  the  coefficients  in  the 
development  of  the  first  member  of  (4),  it  will  be  convenient  to 
arrange  them  in  the  following  manner : 

1st.  The  coefficients  of  constituents  found  in  1 - 0. 

2nd.  The  coefficient  of  i,,  x2  . . x„,  if  found  in  0. 

3rd.  The  coefficients  of  constituents  found  in  0,  excluding  the 
constituent  xx,  x2  . . xn. 

The  above  is  manifestly  an  exhaustive  classification. 

First  then  ; the  coefficient  of  any  constituent  found  in  1 - 0, 
will,  in  the  development  of  the  first  member  of  (4),  be  of  the  form 

1 + positive  terms  derived  from  2. 

Hence,  every  such  coefficient  may  be  replaced  by  unity,  Prop.  i. 
Chap.  ix. 

Secondly  ; the  coefficient  of  xx  . . xn,  if  found  in  0,  in  the 
development  of  the  first  member  of  (4)  will  be 

2sr,  or  h + i2  . . + ln 

Thirdly;  the  coefficient  of  any  other  constituent,  a?!  . . xh 
xi+l  . . xn,  found  in  0,  in  the  development  of  the  first  member 
of  (4)  will  be  ?!  . . + + 5i+1  . . + sn. 

Now  it  is  seen,  that  E is  the  product  of  all  the  coefficients 
above  determined ; but  as  the  coefficients  of  those  constituents 
which  are  not  found  in  0 reduce  to  unity,  E may  be  regarded  as 
the  product  of  the  coefficients  of  those  constituents  which  are  found 
in  0.  From  the  mode  in  which  those  coefficients  are  formed,  we 
derive  the  following  rule  for  the  determination  of  E,  viz.,  in 
each  constituent  found  in  0,  except  the  constituent  xx  x2 . . xn , 
for  xx  write  sx,  for  xx  write  sj,  and  so  on,  and  add  the  results; 
but  for  the  constituent:?!,  x2..xn,  if  it  occur  in0,  write  lx  + s2..+  sn; 
the  product  of  all  these  sums  is  E. 

To  find  E'  we  must  in  (3)  make  tv  = 0,  and  eliminate  xx , x2 . . xn 
from  the  reduced  equation.  That  equation  will  be 

2 [ (xr  4?1..  + Xn)  Sr  + Sr  (xr  - Xx  ...?„)}+  0 = 0.  (5) 


CHAP.  XIX.]  OF  STATISTICAL  CONDITIONS. 


313 


Hence  E'  will  be  formed  from  tbe  constituents  in  1 - 0,  i.  e. 
from  the  constituents  not  found  in  0 in  the  same  way  as  E is 
formed  from  the  constituents  found  in  0. 

Consider  next  the  equation 


This  gives 


Ew  + E'  (1  - w)  - 0. 


E' 


w = 


E-E 


(6) 


Now  E and  E are  functions  of  the  symbols  sx , s2  . . sn . The 
expansion  of  the  value  of  w will,  therefore,  consist  of  all  the  con- 
stituents which  can  be  formed  out  of  those  symbols,  with  their 
proper  coefficients  annexed  to  them,  as  determined  by  the  rule 
of  development. 

Moreover,  E and  E are  each  formed  by  the  multiplication  of 
factors,  and  neither  of  them  can  vanish  unless  some  one  of  the 
factors  of  which  it  is  composed  vanishes.  Again,  any  factor,  as 
?!  . . + Jn  can  only  vanish  when  all  the  terms  by  the  addition  of 
which  it  is  formed  vanish  together,  since  in  development  we  at- 
tribute to  these  terms  the  values  0 and  1 , only.  It  is  further  evi- 
dent, that  no  two  factors  differing  from  each  other  can  vanish 
together.  Thus  the  factors  7X  + s2  • • + 7„,  and  s1  + s2 . . + sn,  can- 
not simultaneously  vanish,  for  the  former  cannot  vanish  unless 
Si  = 0,  or  Si  = 1 ; but  the  latter  cannot  vanish  unless  Si  = 0. 

First,  let  us  determine  the  coefficient  of  the  constituent 
si  7, . : sn  in  the  development  of  the  value  of  w. 

The  simultaneous  assumption  7i  = 1,  s2  = 1 . . Jn  = 1,  would 
cause  the  factor  Si  + s2  . . + sn  to  vanish  if  this  should  occur  in 
E or  E';  and  no  other  factor  under  the  same  assumption  would 
vanish ; but  Si  + s2  . . + sn  does  not  occur  as  a factor  of  either 
E or  E';  neither  of  these  quantities,  therefore,  can  vanish;  and, 

therefore,  the  expression  ^ ^ , is  neither  1 , 0,  nor  ^ . 

Wherefore  the  coefficient  ofsi  s2  . . s„  in  the  expanded  value 
of  io,  may  be  represented  by  ^ . 

Secondly,  let  us  determine  the  coefficient  of  the  constituent 


314 


OF  STATISTICAL  CONDITIONS* 


[CHAF.  XIXo 

The  assumptions  Si=l,s*=l,..«n  = l,  would  cause  the  factor 
h + So  . • + s„  to  vanish.  Now  this  factor  is  found  in  E and  not 
in  E'  whenever  0 contains  both  the  constituents  xx  x2  . . x„.  and 

E'  E' 

a',  x-  . . xn-  Here  then  — = becomes -77,  or  1.  The  factor 

E'  - E E 

Ji  + 7S  . . + ln  is  found  in  E and  not  in  E,  if  <p  contains  neither 
of  the  constituents  x,  x2  . . xn  and  Xi  x2  . . x„.  Here  then 

becomes  or  0.  Lastly,  the  factor  + s2  . . + sn  is 
E — E — E 

contained  in  both  E and  E',  if  one  of  the  constituents  xx  x2 . . x„ 

E' 

and  x\x2 . . xn  is  found  in  0,  and  one  is  not.  Here  then 


E'-E 


becomes 


0 


The  coefficient  of  the  constituent  sl  s2  . . sn,  will  therefore  be 

1,  0,  or  according  as  0 contains  both  the  constituents  xlx2 . . xn 

and  xx  x2  . • xn,  or  neither  of  them , or  one  of  them  and  not  the 
other. 

Lastly,  to  determine  the  coefficient  of  any  other  constituent 

3.S  Si  • • Si  S[+ 1 • . Sn  • 

The  assumptions  sr  - 1,  . . si  = 1,  si+l  = 0,  sn  = 0,  would 
cause  the  factor  ?!  . . + s*  + si+1  . . + sn  to  vanish.  Now  this  fac- 
tor is  found  in  E,  if  the  constituent  x^_  . . xt  xi+1  . . xn  is  found  in 
0 and  in  E\  if  the  said  constituent  is  not  found  in  0.  In  the 
E'  E 

former  case  we  have  77, 77  = 777-  = 1 ; in  the  latter  case  we  have 

E — E E 

. .E..—  = — ° — = 0. 

E'-E  0 -E 

Hence  the  coefficient  of  any  other  constituent  sl . . sh  ~sUl . . ~sn 
is  1 or  0 according  as  the  similar  constituent  x,  . . n xi+1  . . xn 
is  or  is  not  found  in  0. 

We  may,  therefore,  practically  determine  the  value  of  w in 
the  following  manner.  Rejecting  from  the  given  expression  of 
0 the  constituents  xx  x2  . . xn  and  and  x\  x2 . . xn,  should  both  or 
either  of  them  be  contained  in  it,  let  the  symbols  x2,  . . xn, 
in  the  result  be  changed  into  s15  s2,  . . sn  respectively.  Let  the  co- 
efficients of  the  constituents  Sj  s2 . . sn  and  . . ~sn  be  determined 


OF  STATISTICAL  CONDITIONS. 


315 


CHAP.  XIX.] 


according  to  the  special  rules  for  those  cases  given  above,  and  let 
every  other  constituent  have  for  its  coefficient  0.  The  result 
will  be  the  value  of  to  as  a function  of  Sj , s2 , . . s„ . 

As  a particular  case,  let  <p  = xx.  It  is  required  from  the 
given  data  to  determine  the  probability  of  the  event  xx . 

The  symbol  xx,  expanded  in  terms  of  the  entire  series  of  sym- 
bols xx,  x2,  . . xn,  will  generate  all  the  constituents  of  those 
symbols  which  have  xx  as  a factor.  Among  those  constituents 
will  be  found  the  constituent  xx  x2  . . xn , but  not  the  constituent 

X \ X 2 • • Xji  * 

Hence  in  the  expanded  value  of  xx  as  a function  of  the  sym- 
bols Si , s2 , . . s„ , the  constituent  sx  s2 . . sn  will  have  the  coefficient 

0 . .1 

- , and  the  constituent  sx  s2.  . sn  the  coefficient  - . 

U 0 


If  from  xx  we  reject  the  constituent  xx  x2  . . xn,  the  result 
will  be  xx  - xx  x2  . . x„ , and  changing  therein  X\  into  sx , &c.,  we 
have  sx  - sxs2  < . sn  for  the  corresponding  portion  of  the  expres- 
sion of  x\  as  a function  of  sx , s2,  . . sn. 

Hence  the  final  expression  for  xx  is 


0 1 __  _ 

- 5l  “ Sl  S2  . . Sn  + - Si  + - 5i  . . Sn  ^ ^ 

+ constituents  whose  coefficients  are  0. 

The  sum  of  all  the  constituents  in  the  above  expansion  whose 

0 

coefficients  are  either  1,  0,  or  — , will  be  1 - Si$s  • • 

u 

We  shall,  therefore,  have  the  following  algebraic  system  for 
the  determination  of  Prob.  xx,  viz. : 


73  U Sx  — SXS2 

Prob.  xx  = — 


S n 4*  CS\S2 


l -sxs2 


Sn 


with  the  relations 


(8) 


S%  Sn 

Pi  Pz  ‘ ' ~ Pn  (9) 

= 1 — Si  s2  . . sn  = X. 

It  will  be  seen,  that  the  relations  for  the  determination  of 
Si  s3  . . s„  are  quite  independent  of  the  form  of  the  function 
and  the  values  of  these  quantities,  determined  once,  will  serve 


316 


OF  STATISTICAL  CONDITIONS. 


[CHAP.  XIX. 

for  all  possible  problems  in  which  the  data  are  the  same,  how- 
ever the  qucesita  of  those  problems  may  vary.  The  nature  of 
that  event,  or  combination  of  events,  whose  probability  is  sought, 
will  affect  only  the  form  of  the  function  in  which  the  determined 
values  of  s,  s2 . . sn  are  to  be  substituted. 

We  have  from  (9) 

Si  = s3  = p2\,  . . s„  = pn A. 

Whence 

1 - (1  -pi\)  (1  -.p,X)  . . (1  -pn A)  = A. 

Or, 

1 - A = (1  ~PiX)  (1  -p2 A) . . (1  -p„ A) ; (10) 

from  which  equation  the  value  of  A is  to  be  determined. 

Supposing  this  value  determined,  the  value  of  Prob.  xx  will  be 

Pl  A - ( 1 - c)  Pi  p2  . . pn  Xn 
1 - (1  - Pi  A)  (1  - Pi  A)  . . (1  - pn  A)’ 

or,  on  reduction  by  (10), 

Prob. xy  = px  - (1  - c)  pxp2 . .pn\n-1.  (11) 

Let  us  next  seek  the  conditions  which  must  be  fulfilled 
among  the  constants  p{,  p2,  . . pn,  and  the  limits  of  the  value  of 
Prob.  xx . 

As  there  is  but  one  term  with  the  coefficient  - , there  is  but 
one  condition  among  the  constants,  viz., 

Minor  limit,  (1  - s,)  (1  - s2)  . . (1  - sn)  < 0. 

Or,  n (1  - sx)  + n (1  - s2)  . .+n(l  - sn)  - (n-  1)  h(1)<  0. 

Or,  n (1 ) - n (s,)  - n ( s2 ) . . - n (s„)  < 0. 

Whence  P\  + p2  . . + pn  > 

the  condition  required. 

The  major  limit  of  Prob.  xx  is  the  major  limit  of  the  sum  of 
those  constituents  whose  coefficients  are  1 or  - . But  that  sum  is  sx. 
Hence, 

Major  limit,  Prob.  xx  = major  limit  sx  = py . 


CHAP.  XIX.] 


OF  STATISTICAL  CONDITIONS. 


317 


The  minor  limit  of  Prob.  a?i  will  be  identical  with  the  minor 
limit  of  the  expression 

sl  ~ sl  s2  • • sn  + (1  ~ $i)  (1  — S2)  • • (1  — sn')’ 

A little  attention  will  show  that  the  different  aggregates, 
terms  which  can  be  formed  out  of  the  above,  each  including  the 
greatest  possible  number  of  constituents,  will  be  the  following, 
viz. : 

Sl  (1  — S2),  s1  (1  — S3),  . . Sl  (1  — Sn),  (1  — S2)  (1  — S3)  . . (1  — Sn). 

From  these  we  deduce  the  following  expressions  for  the  minor 
limit,  viz. : 

P1-P2,  Pi -Pi  • • P\  ~Pn,  1 - pt  ~p3  • • ~Pn- 
The  value  of  Prob.  xx  will,  therefore,  not  fall  short  of  any  of 
these  values,  nor  exceed  the  value  ofpi. 

Instead,  however,  of  employing  these  conditions,  we  may 
directly  avail  ourselves  of  the  principle  stated  in  the  demon- 
stration of  the  general  method  in  probabilities.  The  condition 
that  Si , s2 , . . sn  must  each  be  less  than  unity,  requires  that  A 

should  be  less  than  each  of  the  quantities  — , . And 

Pi  Pi  Pn 

the  condition  that  Si,  s2,  . . s„,  must  each  be  greater  than  0,  re- 
quires that  A should  also  be  greater  than  0.  Now  px  p2  ■ . pn 
being  proper  fractions  satisfying  the  condition 

Pi  + Pi  • • + Pn  > 1, 

it  may  be  shown  that  but  one  positive  value  of  A can  be  deduced 
from  the  central  equation  (10)  which  shall  be  less  than  each  of 

the  quantities  — , That  value  of  A is,  therefore,  the 

P 1 Pi  Pn 

one  required. 

To  prove  this,  let  us  consider  the  equation 

(1  - j?iA)  (1  - p2X)  • • (1  - Pn\)  - 1 + A = 0. 

When  A = 0 the  first  member  vanishes,  and  the  equation  is 
satisfied.  Let  us  examine  the  variations  of  the  first  member 

between  the  limits  A = 0 and  A = — , supposing  px  the  greatest  of 

Pi 

the  values  j»i  p2  . . pn . 


OF  STATISTICAL  CONDITIONS. 


318 


[chap,  XIX. 


Representing  the  first  member  of  the  equation  by  F,  we  have 


dV 

dX 


— Pi  (f  ,P?.A)  • • PnX~)  . . pn  (1  — PiX ) . . (l  ~/>7j-iA)  + l, 


which,  when  A = 0,  assumes  the  form  - pi  - p2  . . - pn  + 1,  and 
is  negative  in  value. 

Again,  we  have 

d2V 

^7=  J31j02(l-jD3A)(l-JpnA)+&C.» 

consisting  of  a series  of  terms  which,  under  the  given  restrictions 
with  reference  to  the  value  of  A,  are  positive . 

| 

Lastly,  when  A = — , we  have 

Pi 

F = - 1 + 

Pi 

which  is  positive. 

From  all  this  it  appears,  that  if  we  construct  a curve,  the  or- 
dinates of  which  shall  represent  the  value  of  F corresponding  to 
the  abscissa  A,  that  curve  will  pass  through  the  origin,  and  will 
for  small  values  of  A lie  beneath  the  abscissa.  Its  convexity  will, 

I 

between  the  limits  A = 0 and  A = — be  downwards,  and  at  the 

Pi 

i 

extreme  limit  — the  curve  will  be  above  the  abscissa,  its  ordinate 
Pi 

being  positive.  It  follows  from  this  description,  that  it  will  in- 
tersect the  abscissa  once,  and  only  once,  within  the  limits  sped- 

1 

bed,  viz.,  between  the  values  A = 0,  and  A = — . 

Pi 

The  solution  of  the  problem  is,  therefore,  expressed  by  (11), 
the  value  of  A being  that  root  of  the  equation  (10),  which  lies 
, 1 1 1 

within  the  limits  0 and  — , — , . . — . 

Pi  Pi  Pn. 

The  constant  c is  obviously  the  probability,  that  if  the  events 
Xi,  x2,  . . xn,  all  happen,  or  all  fail,  they  will  all  happen. 

This  determination  of  the  value  of  A suffices  for  all  problems 
in  which  the  data  are  the  same  as  in  the  one  just  considered.  It 
is,  as  from  previous  discussions  we  are  prepared  to  expect,  a de- 
termination independent  of  the  form  of  the  function  <p. 


OF  STATISTICAL  CONDITIONS. 


319 


CHAF.  XIX.] 

Let  us.  as  another  example5  suppose 

$ = or  W = Xi  (1  - X2)  . . (1  - Xn)  . . + Xn  (1  - 2a)  . . (1  - Xn.i).  ' 

This  is  equivalent  to  requiring  the  probability,  that  of  the  events 
, x2 , . . xn  one,  and  only  one.  will  happen.  The  value  of  w will 
obviously  be 

l 

W = Si  (^1  — S2) " " (1  — Sn)  • • + Sy;  (I  — 3j)  ■ .(  1 — Sn  _j)  + — . . (1  — Sn)f 

from  which  we  should  have 

Prob.  [xi  (1  -^..(1-*,).,+  xn  (1  -a^)  . . (1  -xn.i)} 

_ Si  (1  — Sz)  . . (1  — Sn)  . • + Sn  (1  — Si)  . . (1  — 3fi-i) 

1 ~{l-Si)..(l~Sn) 

_pi\(l-p2\)  . . (l-j»nA)  ..+pnX(l-pi\)  . . (1—  pn-lX) 

_ _ 

Pi  (1  - A)  p2  (1  - X)  pn  (1  - A) 

1 - piX  1 - p%X  " l-y>„A 

This  solution  serves  well  to  illustrate  the  remarks  made  in  the 
introductory  chapter  (I.  16)  The  essential  difficulties  of  the 
problem  are  founded  in  the  nature  of  its  data  and  not  in  that  of 
its  qusesita.  The  central  equation  by  which  A is  determined,  and 
the  peculiar  discussions  connected  therewith,  are  equally  perti- 
nent to  every  form,  which  that  problem  can  be  made  to  assume, 
by  varying  the  interpretation  of  the  arbitrary  elements  in  its 
original  statement. 


320 


PROBLEMS  ON  CAUSES. 


[CHAr.  XX. 


CHAPTER  XX. 

PROBLEMS  RELATING  TO  THE  CONNEXION  OF  CAUSES  AND 
EFFECTS. 

^0  to  apprehend  in  all  particular  instances  the  relation  of 
cause  and  effect,  as  to  connect  the  two  extremes  in  thought 
according  to  the  order  in  which  they  are  connected  in  nature 
(for  the  modus  operandi  is,  and  must  ever  be,  unknown  to  us), 
is  the  final  object  of  science.  This  treatise  has  shown,  that  there 
is  special  reference  to  such  an  object  in  the  constitution  of  the 
intellectual  faculties.  There  is  a sphere  of  thought  which  com- 
prehends things  only  as  coexistent  parts  of  a universe ; but 
there  is  also  a sphere  of  thought  (Chap,  xi.)  in  which  they  are 
apprehended  as  links  of  an  unbroken,  and,  to  human  appear- 
ance, an  endless  chain — as  having  their  place  in  an  order  con- 
necting them  both  with  that  which  has  gone  before,  and  with 
that  which  shall  follow  after.  In  the  contemplation  of  such 
a series,  it  is  impossible  not  to  feel  the  pre-eminence  which  is  due, 
above  all  other  relations,  to  the  relation  of  cause  and  effect. 

Here  I propose  to  consider,  in  their  abstract  form,  some  pro- 
blems in  which  the  above  relation  is  involved.  There  exists 
among  such  problems,  as  might  be  anticipated  from  the  nature 
of  the  relation  with  which  they  are  concerned,  a wide  diversity. 
From  the  probabilities  of  causes  assigned  a priori , or  given  by 
experience,  and  their  respective  probabilities  of  association  with 
an  effect  contemplated,  it  may  be  required  to  determine  the  pro- 
bability of  that  effect ; and  this  either,  1st,  absolutely,  or  2ndly, 
under  given  conditions.  To  such  an  object  some  of  the  earlier 
of  the  following  problems  relate.  On  the  other  hand,  it  may  be 
required  to  determine  the  probability  of  a particular  cause,  or  of 
some  particular  connexion  among  a system  of  causes,  from  ob- 
served effects,  and  the  known  tendencies  of  the  said  causes,  singly 
or  in  connexion,  to  the  production  of  such  effects.  This  class  of 
questions  will  be  considered  in  a subsequent  portion  of  the 


CHAP.  XX.]  PROBLEMS  ON  CAUSES.  321 

chapter,  and  other  forms  of  the  general  inquiry  will  also  be 
noticed.  I would  remark,  that  although  these  examples  are  de- 
signed chiefly  as  illustrations  of  a method , no  regard  has  been 
paid  to  the  question  of  ease  or  convenience  in  the  application  of 
that  method.  On  the  contrary,  they  have  been  devised,  with 
whatever  success,  as  types  of  the  class  of  problems  which  might 
be  expected  to  arise  from  the  study  of  the  relation  of  cause  and 
effect  in  the  more  complex  of  its  actual  and  Visible  manifestations. 

2.  Problem  I. — The  probabilities  of  two  causes  Ax  andM2 
are  cx  and  c2  respectively.  The  probability  that  if  the  cause  A x 
present  itself,  an  event  E will  accompany  it  (whether  as  a conse- 
quence of  the  cause  A x or  not)  is  px , and  the  probability  that  if 
the  cause  A2  present  itself,  that  event  E will  accompany  it, 
whether  as  a consequence  of  it  or  not,  is  p2 . Moreover,  the 
event  E cannot  appear  in  the  absence  of  both  the  causes  Ax  and 
A2.*  Required  the  probability  of  the  event  E. 

The  solution  of  what  this  problem  becomes  in  the  case  in 
which  the  causes  Ax,  A2  are  mutually  exclusive,  is  well  known 
to  be 

Prob.  E = cxpx  + c2p2; 

and  it  expresses  a particular  case  of  a fundamental  and  very  im- 
portant principle  in  the  received  theory  of  probabilities.  Here 
it  is  proposed  to  solve  the  problem  free  from  the  restriction  above 
stated. 


• The  mode  in  which  such  data  as  the  above  might  be  furnished  by  expe- 
rience is  easily  conceivable.  Opposite  the  window  of  the  room  in  which  I write 
is  a field,  liable  to  be  overflowed  from  two  causes,  distinct,  but  capable  of  being 
combined,  viz.,  floods  from  the  upper  sources  of  the  River  Lee,  and  tides  from 
the  ocean.  Suppose  that  observations  made  on  N separate  occasions  have 
yielded  the  following  results : On  A occasions  the  river  was  swollen  by  freshets, 
and  on  P of  those  occasions  it  was  inundated,  whether  from  this  cause  or  not. 
On  B occasions  the  river  was  swollen  by  the  tide,  and  on  Qof  those  occasions  it 
was  inundated,  whether  from  this  cause  or  not.  Supposing,  then,  that  the  field 
cannot  be  inundated  in  the  absence  of  both  the  causes  above  mentioned,  let  it  be 
required  to  determine  the  total  probability  of  its  inundation. 

Here  the  elements  a,  b,  p,  q of  the  general  problem  represent  the  ratios 
A P J3  Q 
N’  A’  N’  B' 

or  rather  the  values  to  which  those  ratios  approach,  as  the  value  of  N is  indefi- 
nitely increased. 


322 


PROBLEMS  ON  CAUSES. 


[chap.  XX. 


Let  us  represent 

The  cause  Ax  by  x. 

The  cause  A2  by  y. 

The  effect  E by  z. 

Then  we  have  the  following  numerical  data : 

Prob.  x = Ci,  Prob.  y = c2, 

Prob.  xz  = Cipl}  Prob.  yz  = c2p2. 

Again,  it  is  provided  that  if  the  causes  Al}  A2  are  both  ab- 
sent, the  effect  E does  not  occur ; whence  we  have  the  logical 
equation 

(1  - x)  (1  - y)  = v (1  - z). 

Or,  eliminating  v, 

z (1  - x)  (1  - y)  = 0.  (2) 

Now  assume, 

xz  = s,  yz  = t.  (3) 


Then,  reducing  these  equations  (VIII.  7),  and  connecting  the 
result  with  (2), 

xz(l- s)+  s(l-  xz)  + yz(l- 1)  + t(l~yz)+z(l-x)(l-y)  = 0.  (4) 

From  this  equation,  z must  be  determined  as  a developed 
logical  function  of  x,  y,  s,  and  t,  and  its  probability  thence  de- 
duced by  means  of  the  data, 

Prob .x-cly  Prob. y = c2,  Prob .s  = clpi,  Prob.  t = c2p2,  (5) 

Now  developing(4)  with  respect  to  z,  and  putting  x for  1 - x, 
y for  1 = y,  and  so  on,  we  have 

(xs  + sx  + yt  + ty  + xy ) z + (s  + t)  z = 0, 

s+t 

Z + — z 

s + t -xs- sx  -yt  - ty  - xy 

1 _ 1 _ 1 __ 

= stxy  + - stxy  + - stxy  + - stxy 
0 0 0 

1 - - _ 1 1 

+ — stxy  + stxy  + —stxy  + — stxy 

1 _ 1 _ _ __  1 

+ - stxy  + - stxy  + stxy  + - stxy 

+ 07  txy  + 0 7 txy  + 0 stxy  + 0 stxy. 


(6) 


CHAP,  XX.] 


PROBLEMS  ON  CAUSES. 


323 


From  this  result  we  find  (XVII.  17)5 

V=  stxy  + stxy  4 stxy  + Itxy  + Jtxy 
4 Itxy  4 stxy 
= stxy  4 stxy  4 stxy  + st. 

Whence,  passing  from  Logic  to  Algebra,  we  have  the  following 
system  of  equations,  u standing  for  the  probability  sought : 

stxy  4 stxy  4 Jtx  stxy  4 stxy  4 Jty 


_ stxy  4 stxy  stxy  \ stxy 
cifl  c2p2 

stxy  4 stxy  4 Itxy  stxy  4 stxy  4 stxy  4 It  T/ 
u = 1 ~=  ’ 

from  which  we  must  eliminate  s,  t,  x,  y,  and  V. 

Now  if  we  have  any  series  of  equal  fractions,  as 


(?) 


a b c 

-7  = Y?  = “ . . = A5 

a b c 


we  know  that 


la  4 mb  4 nc 


- = A. 


la!  4 mb'  4 nc' 

And  thus  from  the  above  system  of  equations  we  may  deduce 

Itxy  stxy  It  y 
u - clpl  u - c2p2  1 - u 

whence  we  have,  on  equating  the  product  of  the  three  first  mem- 
bers to  the  cube  of  the  last, 

sJ2tt2xxyy 


= V3. 

(u-CipO  (u  - c2p2)  (1  -u) 

Again,  from  the  system  (7)  we  have 

Jtx  sty  stxy 


(8) 


1 — U — Ci  4 cxpx  1 - U - C2  4 c2p2  clp1  4 C2p2  - U 
whence  proceeding  as  before 


= V, 


ss'ztt2xxyy 


(I  - Ci  4 Ci pi  - u)  (1  - c2  4 c2p2  - u)  (c^i  4 c2p,  - v) 


= V3.  (9) 


324 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


Equating  the  values  of  F3  in  (8)  and  (9),  we  have 
(«  - c,  px)  ( n - c2p2)  ( 1 -u) 

= {1  -Cj(l  -px) -u)  { 1 — c2  ( 1 -p2)-u)  (<?! p\  + c2p2  — m), 
which  may  be  more  conveniently  written  in  the  form 
(u-clPl)(u-c2p2)  = { 1 - c,  (1  -pQ-m)  {l-c2(l-p2)-«) 

From  this  equation  the  value  of  u may  be  found.  It  remains 
only  to  determine  which  of  the  roots  must  be  taken  for  this  pur- 
pose. 

3.  It  has  been  shown  (XIX.  12)  that  the  quantity  u,  in 
order  that  it  may  represent  the  probability  required  in  the  above 
case,  must  exceed  each  of  the  quantities  c1pl,  c2p2,  and  fall 
short  of  each  of  the  quantities  1 - c,  (1  - pd),  1 - c2  (1  - p2),  and 
Ci  Pi  + c2p2;  the  condition  among  the  constants,  moreover,  being 
that  the  three  last  quantities  must  individually  exceed  each  of 
the  two  former  ones.  Now  I shall  show  that  these  conditions 
being  satisfied,  the  final  equation  (10)  has  but  one  root  which 
falls  within  the  limits  assigned.  That  root  will  therefore  be  the 
required  value  of  u. 

Let  us  represent  the  lower  limits  cxpL,  c2p2,  by  a , b respec- 
tively, and  the  upper  limits  1 — ( 1 — y?i),  l-c2(l  - p2),  and 

Ci pi  + c2p2,  by  a,  b',  c'  respectively.  Then  the  general  equation 
may  be  expressed  in  the  form 

( u - a)  (u  - b ) (1  - u ) - ( a ' - u)  (p  - u ) (c  - u)  = 0,  (11) 
or  (1  — a - b')  u1  — { ab  - a'b'  + {\-a  - b)c)  u + ab  - add  = 0. 

Representing  the  first  member  of  the  above  equation  by  F,  we 
have 

d2V 
AT  = 

Now  let  us  suppose  a the  highest  of  the  lower  limits  of  m,  a the 
lowest  of  its  higher  limits,  and  trace  the  progress  of  the  values 
of  F between  the  limits  u = a and  u = a'. 

When  u = a,  we  see  from  the  form  of  the  first  member  of  ( 1 1 ) 
that  F is  negative,  and  when  u = a we  see  that  V is  positive. 


CHAP.  XX.]  PROBLEMS  ON  CAUSES.  325 

Between  those  limits  V varies  continuously  without  becoming 

d2V . 

infinite,  and  — - is  ahvays  of  the  same  sign. 

Hence  if  u represent  the  abscissa  V the  ordinate  of  a plane 
curve,  it  is  evident  that  the  curve  will  pass  from  a point  below 
the  axis  of  u corresponding  to  u = a,  to  a point  above  the  axis  of 
u corresponding  to  u = a,  the  curve  remaining  continuous,  and 
having  its  concavity  or  convexity  always  turned  in  the  same  di- 
rection. A little  attention  -will  show  that,  under  these  circum- 
stances, it  must  cut  the  axis  of  u once,  and  only  once. 

Hence  between  the  limits  u = a,  u = a',  there  exists  one  value 
of  u,  and  only  one,  which  satisfies  the  equation  (11).  It  will 
further  appear,  if  in  thought  the  curve  be  traced,  that  the  other 
value  of  u will  be  less  than  a when  the  quantity  1 - a - b'  is  po- 
sitive and  greater  than  any  one  of  the  quantities  a,  b',  d when 
1 - a!  - b'  is  negative.  It  hence  follows  that  in  the  solution  of 
(11)  the  positive  sign  of  the  radical  must  be  taken.  We  thus 
find 

ab  - a'b'  + ( 1 - a - V)  c + J Q_ 

2(1 -o'- ft') 

where  Q=  [ab- a'b’+  (1  - a - b')c)2  - 4(1  - a- b')  (ab- a'b'c). 

4.  The  results  of  this  investigation  may  to  some  extent  be 
verified.  Thus,  it  is  evident  that  the  probability  of  the  event  E 
must  in  general  exceed  the  probability  of  the  concurrence  of  the 
event  E and  the  cause  A1  or  A2.  Hence  we  must  have,  as  the 
solution  indicates, 

u>c1p1,  u>c2p2- 

Again,  it  is  clear  that  the  probability  of  the  effect  E must  in 
general  be  less  than  it  would  be  if  the  causes  Alf  A2  were  mu- 
tually exclusive.  Hence 

u < c1p1  + c2 p2. 

Lastly,  since  the  probability  of  the  failure  of  the  effect  E con- 
curring with  the  presence  of  the  cause  Ax  must,  in  general,  be 
less  than  the  absolute  probability  of  the  failure  of  E,  we  have 

Ci  (1  - pi)  < 1 - u, 

.-.  U < 1 — Cl  (1  - Pi). 


326 


PROBLEMS  ON  CAUSES. 


[chap.  XX. 


Similarly, 

u < 3 — c2  ( 1 - p2). 

And  thus  the  conditions  by  which  the  general  solution  was 
limited  are  confirmed. 

Again,  let  px  = 1,  p2  = 1.  This  is  to  suppose  that  when  either 
of  the  causes  Au  A2  is  present,  the  event  E will  occur.  We  have 
then  a - c1}  b = c2,  a = 1,  b = 1,  c'=  c,  + c2,  and  substituting  in 
(13)  we  get 

_ Cl  c2  - Ci  - c2  - 1 + V { {C\  C2  - Cl  ~ c2  - l)2  + 4 (c,  c2-  Cl  - c2)  j 
“ “ ~2 

= Ci  + c2  - CiC2  on  reduction 

= 1 - (1  - Cl)  (1  - c2). 

Now  this  is  the  known  expression  for  thb  probability  that  one 
cause  at  least  will  be  present,  which,  under  the  circumstances,  is 
evidently  the  probability  of  the  event  E. 

Finally,  let  it  be  supposed  that  Ci  and  c2  are  very  small,  so 
that  their  product  may  be  neglected ; then  the  expression  for  u 
reduces  to  cYpx  + c2p2.  Now  the  smaller  the  probability  of  each 
cause,  the  smaller,  in  a much  higher  degree,  is  the  probability  of 
a conjunction  of  causes.  Ultimately,  therefore,  such  reduction 
continuing,  the  probability  of  the  event  E becomes  the  same  as 
if  the  causes  were  mutually  exclusive. 

I have  dwelt  at  greater  length  upon  this  solution,  because  it 
serves  in  some  respect  as  a model  for  those  which  follow,  some  of 
which,  being  of  a more  complex  character,  might,  without  such 
preparation,  appear  difficult. 

5.  Problem  II. — In  place  of  the  supposition  adopted  in  the 
previous  problem,  that  the  event  E cannot  happen  when  both  the 
causes  A j,  A2  are  absent,  let  it  be  assumed  that  the  causes  Au  A2 
cannot  both  be  absent,  and  let  the  other  circumstances  remain  as 
before.  Required,  then,  the  probability  of  the  event  E. 

Here,  in  place  of  the  equation  (2)  of  the  previous  solution,  we 
have  the  equation 

(l-.r)  (l-y)  = 0. 

The  developed  logical  expression  of  z is  found  to  be 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


327 


z - stxy  + - stxy  + - stxy  + -"stxy 
0 0 0 1 

1 _ _ . I __  1 

+ -stxy  + stxy  + - stxy  + - stxy 
0 '0  0 

1_  1 _ . __  1 

+ 0 stxy  + - stxy  + stxy  + - stxy 


1 

+ 0 stxy  + 0 s txy  + 0 s t xy  + - s txy 


and  the  final  solution  is 


Prob.  E = u: 


the  quantity  u being  determined  by  the  solution  of  the  equation 

( u -a)  (u-b)  _ (a!  - u)  (b  -u)  , . 

a + b - u u - a - b + Is  ''  J 

■wherein  a - cxpx,  b = c.2p2,  a = 1 - (1  - pi),  V = 1 - c2  (1  - p). 

The  conditions  of  limitation  are  the  following  That  value 
of  u must  be  chosen  which  exceeds  each  of  the  three  quantities 

a,  b3  and  a + b1  - 1 , 

and  which  at  the  same  time  falls  short  of  each  of  the  three  quan- 
tities 

a',  b.  and  a + b. 


Exactly  as  in  the  solution  of  the  previous  problem,  it  may  be 
shown  that  the  quadratic  equation  (1)  will  have  one  root,  and 
only  one  root,  satisfying  these  conditions.  The  conditions  them- 
selves were  deduced  by  the  same  rule  as  before,  excepting  that 
the  minor  limit  a'  + b - I was  found  by  seeking  the  major  limit 
of  1 - z. 

It  may  be  added  that  the  constants  in  the  data,  beside  satis- 
fying the  conditions  implied  above,  viz.,  that  the  quantities  a,  b, 
and  a + b,  must  individually  exceed  a,  b,  and  a + b - 1,  must 
also  satisfy  the  condition  cx  + c2  > 1.  This  also  appears  from  the 
application  of  the  rule. 

6.  Problem  III. — The  probabilities  of  two  events  A and  B 
are  a and  b respectively,  the  probability  that  if  the  event  A take 
place  an  event  E will  accompany  it  is  p,  and  the  probability  that 


328 


PROBLEMS  ON  CAUSES. 


[chap.  XX 


if  the  event  B take  place,  the  same  event  E will  accompany  it 
is  q.  Required  the  probability  that  if  the  event  A take  place  the 
event  B will  take  place,  or  vice  versa,  the  probability  that  if  B 
take  place,  A will  take  place. 

Let  us  represent  the  event  A by  x,  the  event  B by  y,  and  the 
event  E by  z.  Then  the  data  are — 

Prob.  x = a,  Prob.  y = b. 

Prob.  xz  = ap,  Prob.  yz  = bq. 

Whence  it  is  required  to  find 


Prob.  xy  Prob.  xy 
Prob.®  Prob.y’ 

Let  xy  = s,  yz  - t,  xy  = w. 

Eliminating  z,  we  have,  on  reduction, 


sx  + ty  + syt  + xts  + xyw  + (1  - xy)  w = 0, 

sx  + ty  + syl+  xts  + xy 
’ ’ W 2xy  - 1 

1 _ 1 _ 1 __ 

= xyst  -t  — xyst  + - xyst  + - xyst 

1 _ _ __  1 _ _ 1 

+ - xyst  + 0 xyst  + - xyst+  - xyst 

+ ^ xylt  + ^ xyst  + 0 xy~st  + ^ xyst 

+ xyst  + QxyJI  + OxyHl  + OxysJ.  (1) 

Hence,  passing  from  Logic  to  Algebra, 

„ , xyst  + xyJl 

Prob.  xy  = — y~^’ 


x,  y,  s,  and  t being  determined  by  the  system  of  equations 

xyst  + xysl  + xylT  + xy  Jt  xyst  + xyHt  + xyll  + xyJT 
a b 

xyst  + xysl  xyst  + xylt 
ap  bq 

= xyst  + xysl  + xylt  + xyll  + xyst  + xyll  + xy! t = V. 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


329 


To  reduce  the  above  system  to  a more  convenient  form,  let  every 
member  be  divided  by  xy  77,  and  in  the  result  let 


xs  yt  , x y 

— = m,  — = m,  = = ft,  ^ = 77 . 

xs  yt  x y 


We  then  find 


mm  + m + nn  + n mm  + m + nn  + n 


a b 

mm  + 7n  mm  + m! 


Also, 


ap  bq 

= mm!  + m + m!  + nri+n  + n!  + 1 . 

mm!  + nn 


Prob.  xy  = ; r— — ; 7 — =-. 

J mm  + 777  + 777+7777+77  + 77  + 1 

These  equations  may  be  reduced  to  the  form 

mm!  + m mm!  + m!  nn  + n nn  + ri 

ap  bq  a(\-p)  b{\-q) 

= (to  + 1)  (m!  + 1)  + (ft+  1)  (ft'+  1)  - 1. 

mm'  + nn! 


Prob.  xy  = 


(m  + 1)  (m  + 1)  + (w  + 1)  (w'  + 1 ) - I" 


Now  assume 


(m  + 1)  (771' + 1)  = — — (ft  + 1)  (ft'+  1)  = 


v + y - 1 

7ft  (m!  + 1)  ( 777  + 1) 


v + p ■ 
m/x 


T (2) 


Then  since  mm  + m - — . , 

777+1  (»7  + 1)  (V  + /JL  - 1) 

and  so  on  for  the  other  numerators  of  the  system,  we  find,  on 

substituting  and  multiplying  each  member  of  the  system  by 

v + y - 1,  the  following  results  : 


my 


my 


nv 


ft  V 


= 1. 


(7ft+l)a/>  (m  + l)bq  (ft  + 1)  a C1  _ P)  (»'+ 1)6(1  -q) 

Prob. xy  = (mm  + nn)  (v  + y - 1).  (3) 

Prom  the  above  system  we  have 


777 


777  + 1 y 


= — , whence  m = 


ap 

V-  ~ aP' 


330 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


Similarly 


m = 


p - w 


n = 


Hence  , u 

m + 1 = 


a(l  ~P) 
v - a (1  - p)’ 

n + 1 = 


n = 


p-  ap  v - a ( 1 - p) 

Substitute  these  values  in  (2)  reduced  to  the  form 

p v 


b(l~q) 
v-b(\  -qy 

, &c. 


V + fl-l  = 


(m+l)(m'+l)  (n+l)(n+l)’ 


and  we  have 


v + p - 1 = 


Q*-gp)  (m ~ bl)  (v-a(l-p)}  {v-ft(l-y)} 


P V 

Substitute  also  for  m,  m\  &c.  their  values  in  (3),  and  we  have 

Prob.  xy 

abpq  ab{  1 - p ) (1  - q) 


(4) 


= r : 

Up  - a\ 


(ji  - ap)  (p  - bq)  {v-a(l-p))  {v-b(l-q)} 

= aJm  j ab(l -P)  0 ~g)  by  (4)> 

y i/ 

Now  the  first  equation  of  the  system  (4)  gives 

i z.  aiM 

v + p-  l=  p-ap~bq  + 


(«) 


abPi  ^ j. 

= v - 1 + ap  + bq. 


Similarly, 


«&(i -p)  0 - q) 


= p-  1 + 0(1  -p)+  6(1  -q). 


Adding  these  equations  together,  and  observing  that  the  first 
member  of  the  result  becomes  identical  with  the  expression  just 
found  for  Prob.  xy , we  have 

Prob.  xy  = v + p + a+  b-2. 

Let  us  represent  Prob.  xy  by  m,  and  let  a + b - 2 = m,  then 

(i  + v = u - m.  (6) 

Again,  from  (5)  we  have 

pv  = abpq  - (ap  + bq  - \ ) p. 


(7) 


PROBLEMS  ON  CAUSES. 


331 


CHAP.  XX.] 

Similarly  from  the  first  and  third  members  of  (4)  equated  ayc 
have 

fiv  = ab(  1 -p)  (1  - q)  - {a  (1  - p)  + b(l  - q)  - 1)  Vo 

Let  us  represent  ap  + bq  - 1 by  h,  and  a (1  - p)  + b (1  - q)  - 1 by 
h'.  We  find  on  equating  the  above  values  of  pv, 

bp  - h'v  = ab  [pq  + (1  - p)  (1  - q) ) 

= ab(p  + q - 1). 

Let  ab(p  + q - 1 ) = /,  then 

bp  — h'v  = /.  (8) 

Now  from  (6)  and  (8)  we  get 

h'  (u  -m)  + l h (u-rri)  - l 

p = . v - . 

m m 

Substitute  these  values  in  (7)  reduced  to  the  form 


and  we  have 


p (v  + h ) = abpq, 

(Jiu  - l)  [h‘  (u  - m)  + l } = abpqm 


(9) 


a quadratic  equation,  the  solution  of  Avliich  determines  u,  the  va- 
lue of  Prob.  xy  sought. 

The  solution  may  readily  be  put  in  the  form 

^ lb!  + b(li'm  - l)  ± \J  \_{lb!  - h ( Jim  - l)  }4  + 4hb'abpqm2^\ 

p = _ • 


But  if  we  further  observe  that 

lb'  - h ( h'm  -1)  - l (Ji  + h')  - hli'm  = {l-  bh')  m; 
since  h = ap  + bq  - 1,  h'  = a (1  - p)  + b (I  - q)  - 1, 
whence  h + h'=a  + b-  2 = m, 

we  find  , 

tj  i ™ 7A'  + h(hm  -l)  + m yj  ( (7 - hli)2  + 4 hh'abpq } 

Prob,  xy  = 


(10) 


It  remains  to  determine  which  sign  must  be  given  to  the  radi- 
cal. We  might  ascertain  this  by  the  general  method  exemplified 
in  the  last  problem,  but  it  is  far  easier,  and  it  fully  suffices  in  the 
present  instance,  to  determine  the  sign  by  a comparison  of  the 


332 


PROBLEMS  ON  CAUSES. 


[chap.  XX. 


above  formula  with  the  result  proper  to  some  known  case.  For 
instance,  if  it  were  certain  that  the  event  A is  always , and  the 
event  B never , associated  with  the  event  E,  then  it  is  certain  that 
the  events  A and  B are  never  conjoined.  Hence  if  p = 1,  q = 0, 
we  ought  to  have  u = 0.  Now  the  assumptions  p=  1,  q = 0, 
give 

h-a-  1,  k = b ~ 1,  1=0,  m=a+b-  2. 

Substituting  in  (10)  we  have 


Prob.  xy 


(a  - 1)  b - 1)  (a  + b - 2)  + (a  + b - 2)  (a  - 1)  (b  - 1) 
2 (a  - 1)  (b  - 1) 


and  this  expression  vanishes  when  the  lower  sign  is  taken. 
Hence  the  final  solution  of  the  general  problem  will  be  expressed 
in  the  form 

Prob.a;i/  Ui  + h ( h'm  - l)  - rrnj  {(l  - hti)2  + 4 hh'abpq) 
Prob.  x ‘iahh ' 


wherein  h = ap  + bq  - 1,  h'  = a (1  - p)  + b (1  - q)  - 1, 
l = ab(p  + q - 1),  m = a + b - 2. 

As  the  terms  in  the  final  logical  solution  affected  by  the  co- 
efficient ^ are  the  same  as  in  the  first  problem  of  this  chapter, 
the  conditions  among  the  constants  will  be  the  same,  viz., 
ap  < 1 -b  (1  - q),  bq<\-a(\-p). 

7.  It  is  a confirmation  of  the  correctness  of  the  above  solution 
that  the  expression  obtained  is  symmetrical  with  respect  to  the 
two  sets  of  quantities  p,  q,  and  1 -p,  1 - q,  i.  e.  that  on  changing 
p into  1 - p,  and  q into  1 - q,  the  expression  is  unaltered  This 
is  apparent  from  the  equation 

Prob.  xy  = ab  {«  + 0rP)Q-i>  \ 
t p v J 


employed  in  deducing  the  final  result.  Now  if  there  exist  pro- 
babilities p,  q of  the  event  E,  as  consequent  upon  a knowledge 
of  the  occurrences  of  A and  B,  there  exist  probabilities  1 - p,  1 - q 
of  the  contrary  event,  that  is,  of  the  non-occurrence  of  E under 
the  same  circumstances.  As  then  the  data  are  unchanged  in 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


333 


form,  whether  we  take  account  in  them  of  the  occurrence  or  of 
the  non-occurrence  of  E,  it  is  evident  that  the  solution  ought  to 
be,  as  it  is,  a symmetrical  function  of  p,  q and  1 - p,  1 - q. 

Let  us  examine  the  particular  case  in  which  p - 1 , <7  = 1. 
We  find 

k = a + b - 1,  hi  = - 1,  l = ab,  m = a 4 b - 2, 
and  substituting 

Prob.  xy  _ —ab  + (a  + 6-1)  (2  - a - b- ab)-(a  + b-  2)  (a.b-a-b  + 1) 
Prob.  x - 2a  (a  + b - 1) 

- 2 ab(a  + 6-1) 

- 2a  (a  + b - 1) 

It  would  appear,  then,  that  in  this  case  the  events  A and  B are 
virtually  independent  of  each  other.  The  supposition  of  their 
invariable  association  with  some  other  event  E,  of  the  frequency 
of  whose  occurrence,  except  as  it  may  be  inferred  from  this  par- 
ticular connexion,  absolutely  nothing  is  known,  does  not  establish 
any  dependence  between  the  events  A and  B themselves.  I ap- 
prehend that  this  conclusion  is  agreeable  to  reason,  though  par- 
ticular examples  may  appear  at  first  sight  to  indicate  a different 
result.  For  instance,  if  the  probabilities  of  the  casting  up,  1st, 
of  a particular  species  of  weed,  2ndly,  of  a certain  description  of 
zoophytes  upon  the  sea-shore,  had  been  separately  determined, 
and  if  it  had  also  been  ascertained  that  neither  of  these  events 
could  happen  except  during  the  agitation  of  the  waves  caused  by 
a tempest,  it  would,  I think,  justly  be  concluded  that  the  events 
in  question  were  not  independent.  The  picking  up  of  a piece  of 
seaweed  of  the  kind  supposed  would,  it  is  presumed,  render  more 
probable  the  discovery  of  the  zoophytes  than  it  would  otherwise 
have  been.  But  I apprehend  that  this  fact  is  due  to  our  know- 
ledge of  another  circumstance  not  implied  in  the  actual  conditions 
of  the  problem,  viz.,  that  the  occurrence  of  a tempest  is  but  an 
occasional  phenomenon.  Let  the  range  of  observation  be  con- 
fined to  a sea  ahoays  vexed  with  storm.  It  would  then,  I sup- 
pose, be  seen  that  the  casting  up  of  the  weeds  and  of  the 
zoophytes  ought  to  be  regarded  as  independent  events.  Now, 
to  speak  more  generally,  there  are  conditions  common  to  all  phte- 


334 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


nomena, — conditions  which,  it  is  felt,  do  not  affect  their  mutual 
independence.  I apprehend  therefore  that  the  solution  indicates, 
that  when  a particular  condition  has  prevailed  through  the  whole 
of  our  recorded  experience,  it  assumes  the  above  character  with 
reference  to  the  class  of  phenomena  over  which  that  experience 
has  extended. 

8.  Problem  IV. — To  illustrate  in  some  degree  the  above 
observations,  let  there  be  given,  in  addition  to  the  data  of  the 
last  problem,  the  absolute  probability  of  the  event  E,  the  com- 
pleted system  of  data  being 

Prob.  x=  a,  Prob.  y = b,  Prob.  z = c, 

Prob.  xz  = ap,  Prob.  yz  = bq, 


and  let  it  be  required  to  find  Prob.  xy. 

Assuming,  as  before,  xz  = s,  yz  = t,  xy  - w,  the  final  logical 
equation  is 

w = xystz  + xysTz  + 0 ( xystz  + xystz  + xyz~s7  + xyzsl 

+ xyzJT  + xyzsT). 

+ terms  whose  coefficients  are  (1) 


The  algebraic  system  having  been  formed,  the  subsequent  elimi- 
nations may  be  simplified  by  the  transformations  adopted  in  the 
previous  problem.  The  final  result  is 

Prob.  xy  - ab  (2) 

The  conditions  among  the  constants  are 

c > ap,  c > bq,  c < 1 - a (1  - p),  c <1  - b (1  - q). 

Now  ifp  = 1,  q = 1,  we  find 

Prob.  xy  = 

c not  admitting  of  any  value  less  than  a or  b.  It  follows  hence 
that  if  the  event  E is  known  to  be  an  occasional  one,  its  inva- 
riable attendance  on  the  events  x and  y increases  the  probability 
of  their  conjunction  in  the  inverse  ratio  of  its  own  frequency. 


PROBLEMS  ON  CAUSES. 


335 


CHAP.  XX.] 


The  formula  (2)  may  be  verified  in  a large  number  of  cases. 
As  a particular  instance,  let  q - c,  we  find 

Prob.  xy  = ab.  (3) 

Now  the  assumption  q = c involves,  by  Definition  (Chap.  XVI.) 
the  independence  of  the  events  B and  E.  If  then  B and  E are 
independent,  no  relation  which  may  exist  between  A and  E can 
establish  a relation  between  A and  B ; wherefore-  A and  B are 
also  independent,  as  the  above  equation  (3)  implies. 

It  may  readily  be  shown  from  (2)  that  the  value  of  Prob.  z, 
which  renders  Prob.  xy  a minimum,  is 


Prob.  z = 


V(P9) 

V(pg ) + V(i  -p)  (i  - q)' 


If  p = q,  thi§  gives 


Prob.  z = p\ 


a result,  the  correctness  of  which  may  be  shown  by  the  same  con- 
siderations which  have  been  applied  to  (3). 

Problem  V. — Given  the  probabilities  of  any  three  events, 
and  the  probability  of  their  conjunction ; required  the  proba- 
bility of  the  conjunction  of  any  two  of  them. 

Suppose  the  data  to  be 

Prob.  x = p,  Prob.  y - q,  Prob.  z = r,  Prob.  xyz  - m, 


and  the  quaesitum  to  be  Prob.  xy. 

Assuming  xyz  = s,  xy  - t,  we  find  as  the  final  logical  equa- 
tion, 

t = xyzs  + xyz!  + 0 (xy!  + x!)  + ^ (sum  of  all  other  constituents)  ; 
whence,  finally, 

Prob.  xy  . H ~ ^ ^ - ipjr*  - 4fyF-), 


wherein  p - 1 -p,  &c.  H=  pq  + (p  + q)r. 

This  admits  of  verification  when  p = 1,  when  <7=1,  when  r = 0, 
and  therefore  m = 0,  &c. 

Had  the  condition,  Prob.  z = r,  been  omitted,  the  solution 
would  still  have  been  definite.  We  should  have  had 


336 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 

Prob.  xy  , 

* 1 - m 

and  it  may  be  added,  as  a final  confirmation  of  their  correctness, 
that  the  above  results  become  identical  when  m = pqr. 

9.  The  following  problem  is  a generalization  of  Problem  I., 
and  its  solution,  though  necessarily  more  complex,  is  obtained  by 
a similar  analysis. 

Problem  VI. — If  an  event  can  only  happen  as  a conse- 
quence of  one  or  more  of  certain  causes  Ax,  At,  ..  A„,  and  if 
generally  c-L  represent  the  probability  of  the  cause  At,  and  pt  the 
probability  that  if  the  cause  Ax  exist,  the  event  E will  occur, 
then  the  series  of  values  of  ct  and  pi  being  given,  required  the 
probability  of  the  event  E * 

Let  the  causes  Ax,  A2, . . A„  be  represented  by  xx,  x2, . .x„, 
and  the  event  E by  z. 

Then  we  have  generally, 

Prob.  xi  = Prob.  xxz  = ctpi. 

Further,  the  condition  that  E can  only  happen  in  connexion  with 
some  one  or  more  of  the  causes  Ax,  A2, . . An  establishes  the  logi- 
cal condition, 

z{\  - xx)  (1  -x2)  . . (1  - xn)  = 0.  (1) 


* It  may  be  proper  to  remark,  that  the  above  problem  was  proposed  to  the 
notice  of  mathematicians  by  the  author  in  the  Cambridge  and  Dublin  Mathema- 
tical Journal,  Nov.  1851,  accompanied  by  the  subjoined  observations  : 

“ The  motives  which  have  led  me,  after  much  consideration,  to  adopt,  with 
reference  to  this  question,  a course  unusual  in  the  present  day,  and  not  upon 
slight  grounds  to  be  revived,  are  the  following : — First,  I propose  the  question 
as  a test  of  the  sufficiency  of  received  methods.  Secondly,  I anticipate  that  its 
discussion  will  in  some  measure  add  to  our  knowledge  of  an  important  branch 
of  pure  analysis.  However,  it  is  upon  the  former  of  these  grounds  alone  that  I 
desire  to  rest  my  apology. 

“ While  hoping  that  some  may  be  found  who,  without  departing  from  the  line 
of  their  previous  studies,  may  deem  this  question  worthy  of  their  attention,  I 
wholly  disclaim  the  notion  of  its  being  offered  as  a trial  of  personal  skill  or 
knowledge,  but  desire  that  it  may  be  viewed  solely  with  reference  to  those  pub- 
lic and  scientific  ends  for  the  sake  of  which  alone  it  is  proposed.” 

The  author  thinks  it  right  to  add,  that  the  publication  of  the  above  problem 
led  to  some  interesting  private  correspondence,  but  did  not  elicit  a solution. 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


337 


Now  let  us  assume  generally 

Xi  z = ti, 

which  is  reducible  to  the  form 

Xiz(  1 - ti)  + ti(  1 - xtz)  = 0, 

forming  the  type  of  a system  of  n equations  which,  together  with 
(1),  express  the  logical  conditions  of  the  problem.  Adding  all 
these  equations  together,  as  after  the  previous  reduction  we  are 
permitted  to  do,  we  have 

S [xiZ(l  -ti)  + ti(l-Xiz)}+z(l  - x 0 (1  -x2)..(l  -xn)  = 0,  (2) 

(the  summation  implied  by  2 extending  from  i=  1 to  i-n),  and 
this  single  and  sufficient  logical  equation,  together  with  the  2 n 
data,  represented  by  the  general  equations 

Prob.  Xi  = ci,  Prob.  tt  = £■;/>;,  (3) 

constitute  the  elements  from  which  we  are  to  determine  Prob.  z. 
Let  (2)  be  developed  with  respect  to  z.  We  have 

[S{X;(1  - ti)  + £;(1  - Xi)}  + (1  - x^  (1  - x2)  . .(1  -xn)]  z 

+ 2ti(l-z)  = 0, 

whence 

™ - (4) 

2/j-  2 {aJi(l  - ti)  + ti(l  - xi))  - (1  - XJ  (1  - x2)  . . (1  -x„)  v ' 

Now  any  constituent  in  the  expansion  of  the  second  member  of 
the  above  equation  will  consist  of  2 n factors,  of  which  n are  taken 
out  of  the  set  x1}  x-2, . . xn,  1 - x1?  1 - x2, . . 1 - xn,  and  n out  of 
the  set  £„  t2, . .tn,  1 - t^  1 - t2, . . 1 - tn,  no  such  combination  as 
Xi  (1  - xx),  #,(1  - t^,  being  admissible.  Let  us  consider  first 
those  constituents  of  which  (1  - ti),  (1  - t2) . . (1  - tn)  forms  the 
/-factor,  that  is  the  factor  derived  from  the  set  /„  . . 1 - tx. 

The  coefficient  of  any  such  constituent  will  be  found  by 
changing  t2,  . . tn  respectively  into  0 in  the  second  member  of 
(4),  and  then  assigning  to  x2,  x2,  . . xn  their  values  as  dependent 
upon  the  nature  of  the  x-factor  of  the  constituent.  Now  simply 
substituting  for  /,,  t2, . . tn  the  value  0,  the  second  member  be- 
comes 

0 

- - (1  - x^  (1  - x2)  . . ( 1 - xny 


338 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 

and  this  vanishes  whatever  values,  0,  1,  we  subsequently  assign  . 
to  xu  x2, . . xn.  For  if  those  values  are  not  all  equal  to  0,  the 
term  does  not  vanish,  and  if  they  are  all  equal  to  0,  the  term 
-(1  — a;,) . . (1  - x„)  becomes  - 1,  so  that  in  either  case  the  denomi- 
nator does  not  vanish,  and  therefore  the  fraction  does.  Hence 
the  coefficients  of  all  constituents  of  which  (1  - #i)  . . (1  - tn)  is  a 
factor  will  be  0,  and  as  the  sum  of  all  possible  ^-constituents  is 
unity,  there  will  be  an  aggregate  term  0(1-^)  . . (1  - t„)  in  the 
development  of  z . 

Consider,  in  the  next  place,  any  constituent  of  which  the 
^-factor  is  tx  t2 . . tr  ( l - £r+1) . . (1  - tn),  r being  equal  to  or  greater 
than  unity.  Making  in  the  second  member  of  (4),  = 1,  1, 

tr+ 1 = 0, . . tn  = 0,  we  get  the  expression 

r 

+ xr  - xM . . - xn  - (i  - Xj)  (i  - x2) ..  (I  - xny 

Now  the  only  admissible  values  of  the  symbols  being  0 and  1, 
it  is  evident  that  the  above  expression  will  be  equal  to  1 when 
xi  = 1 . . xr  = 1,  xM  = 0, . . xn  = 0,  and  that  for  all  other  combi- 
nations of  value  that  expression  will  assume  a value  greater  than 
unity.  Hence  the  coefficient  1 will  be  applied  to  all  constituents 
of  the  final  development  which  are  of  the  form 

X\  • . Xr  [1  — Xr+\)  . • [1  X n)  t\  . . tr  [1  tr+ 1)  • • [1  ~ 

the  ^-factor  being  similar  to  the  ^-factor,  while  other  consti- 
tuents included  under  the  present  case  will  have  the  virtual  co- 
efficient Also,  it  is  manifest  that  this  reasoning  is  independent 

of  the  particular  arrangement  and  succession  of  the  individual 
symbols. 

Hence  the  complete  expansion  of  z will  be  of  the  form 
2 = S (XT)  + 0 (1  - *,)  (1  - t2)  . . (i  - t„ ) 

+ constituents  whose  coefficients  are  -,  (5) 

where  T represents  any  ^-constituent  except  (1  (1  - tn), 

and  X the  corresponding  or  similar  constituent  of  x, . . x„. 


PROBLEMS  ON  CAUSES. 


339 


CHAP.  XX.] 


For  instance,  if  n = 2,  we  shall  have 

'J.  ] — tP]  tx  1 Xj  X2  ^1  ^2  4"  Xj  2/2  ^2J 
x,,  x2,  &c.  standing  for  1 - Xj,  1 - X2,  &c. ; whence 

Z — 0C\  t\  ts£  “1"  0C\  X2  t%  "l-  X\  X‘2  t\  t-2 

+ 0 (Xy  X-2  lx  12  + X , X2  1\  1-2  + X\  %2  lx  1 + Xl  li  1 Tl)  ffi'l 
+ constituents  whose  coefficients  are  -. 


This  result  agrees,  difference  of  notation  being  allowed  for,  with 
the  developed  form  of  z in  Problem  I.  of  this  chapter,  as  it  evi- 
dently ought  to  do. 

10.  To  avoid  complexity,  I purpose  to  deduce  from  the  above 
equation  (6)  the  necessary  conditions  for  the  determination  of 
Prob.  z for  the  particular  case  in  which  n = 2,  in  such  a form  as 
may  enable  us,  by  pursuing  in  thought  the  same  line  of  investi- 
gation, to  assign  the  corresponding  conditions  for  the  more  gene- 
ral case  in  which  n possesses  any  integral  value  whatever. 

Supposing  then  n = 2,  we  have 


V — Xx  X2  tx  1 4"  &X  X2  H 4*  Xx  X2  1 1 4~  Xi  X2  1 t 2 4“  X)  X2  tx  1 
+ Xx  X2  t ] t2  4“  X,  X2  tx  t2» 


Prob.  z = 


Xx  X2  tx  t2  4"  Xj  X2  tx  1 2 "4"  Xy  X2  t\t2 


V 


» 


the  conditions  for  the  determination  of  x19  tx,  &c.,  being 

Xx  x2  tx  t2  + Xx  x2  txl2  + x,  x2 1x1  + xx  x2  7]  1 
Cl 

Xx  X2  tx  t2  4"  Xj  X2  tx  t2  4"  X]  X2  tx  t2  4 Xj  X2  tx  t2 
C2 

Xj  X2  tx  t2  4-  X|  X2  tx  t2  Xx  X2  1 1 to  + Xx  X2  tx  1 2 y -r 

CxPx  c2p2 

Divide  the  members  of  this  system  of  equations  by  l x2 11, 
and  the  numerator  and  denominator  of  Prob.  z by  the  same  quan- 
tity, and  in  the  results  assume 

Xx  t\  X2  t2  Xx  X2 

= W, , — r=T  = ffl.,  — = «i,  — = 

Xx  tx  x2  t2  Xx  X2 


(7) 


340 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


we  find 


Prob.  z = 


to,to2  + to,  + m2 


and 


to,to2  + ml  + m%  + nxn%  + nx  + n%  + l’ 

W*,TO2  + TO,  + 71x71%  + Tlx  771x771%  + 711%  + 11x71%  + 71% 


c% 


771x771%  + TTlx  771x771%  + 771% 


= 771x771%  + m,  + 771%  + 71x71%  + 7lx  + 71%  + 1 , (8) 


Clfl  c%p% 

whence,  if  we  assume, 

(ttix  + 1)  (m%  + 1)  = M,  (nx  + 1)  ( ti%  + 1)  = N,  (9) 
we  have,  after  a slight  reduction, 


Prob. z = — 


M-  1 


M + N-  1’ 


»1  (n%  + 1)  = 7l%(7lx  + 1)  = TTlx  (771%  ~t  1)  _ 771%  (to,  + 1)  _ „ XT_  i . 
c2(l-p2)  c,p,  c2p2 


or, 


TtlxM 


M 


7lxN 


(jrix  + 1)  cxpx  (. m%+l)c%p%  (nx  + 1)  Cx  (l  - px) 

n%N 


(n%+  1)  c2  ( 1 - p% ) 


= M+  N-  1. 


Now  let  a similar  series  of  transformations  and  reductions  be 
performed  in  thought  upon  the  final  logical  equation  (5).  We 
shall  obtain  for  the  determination  of  Prob.  z the  following  ex- 
pression : 


Prob.  z = 


M - 1 


wherein 


m. 


M+N- l9 

M = (TO,  + 1)  (771%  + 1)  . . (TO„+  1), 

N = (nx  + 1)  (n%  + 1 )..(«„+  1), 
to„,  nx,  . . nn,  being  given  by  the  system  of  equations, 

ttixM  m%M  to„M 

(to!  + 1)  Cxpx  (m%  + 1)  c%p%  ' ' (to„  + 1)  cnpn 

tixN  nnN 


(10) 


(Wi+  1)^(1  -px)  ‘ ' (nn+  1)  cn(l  - pn ) 
Still  further  to  simplify  the  results,  assume 


= M + N-  1. 


(11) 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


341 


whence 


We  find 


M+N  - 11  M + N-l  _ 1 
M fj. ’ N v 


M - 


N = 


fi  + v — 1 JU  + v - 1 


772, 


m„ 


(772,  + l)c,/7,  (m2  + 1)  c2p2  (mn  + 1)  cnpn  ft 


77, 


772 


Mn 


_ 1 

(77,+  1)  C,  (1  -/>,)  (772  + 1)C2(1  -p2)'  ' (ltn  + l)cn(l-pn)  v’ 

whence 


C\P\  CnPn 

777 1 = , ..  777n  = 


and  finally, 


M 


b~Cnpn 


, U fl 

777,  + 1 = — — , . . 77?„  + 1 = 


77,  + 1 = 


/K  - C,/7, 
V 

V-C j (1-/7,) 


, . . 7?„  + 1 — 


^ C-nPn 

V 


v-cn(\-pny 


Substitute  these  values  with  those  of  ilf  and  N in  (9),  and 
we  have 

= fi 

(m  - C,/7,)  (fl  - C2p2 ) . . 0 - c„/7„)  JU  + V - 1 ’ 

vn  V 


{v  - c,  (1-/7,)}  |v  - C2  (1  -/7*)}  . • {v-  cn  (1  -pn))  fl  + V - 1 

which  may  be  reduced  to  the  symmetrical  form 


M + 


, (fl  - Cl/7,)  • . (fl  - Cnpn ) 
v - 1 = 


Finally, 


[v  - c,  ( 1 - />,)}  . . {v  - c„  (1  - p„)j 


^ u M - 1 

Jrrob.  z = , - — — — - = 1 - v. 


M+N-l 

Let  us  then  assume  1 - v = u,  we  have  then 

(fl  - C,  pi)  . . (fl  — Cn  Pn) 


(12) 

(13) 


- u = 


P 


{1  -c,  (1  -px)  - u)  ..  {1  - c„(l  -pn)-u) 
(1  - u)’1'1 


342 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


If  we  make  for  simplicity 

= «n  cnpn  = an,  1 -cx  (1  -px)  = bu  &c., 
the  above  equations  may  be  written  as  follows : 


wherein 


..  0 ~ax)  . . (p-  an ) 
M pn'1 

(14) 

(bx  - u)  . . (bn  - u) 
^ (1  - u)n~l 

(15) 

This  value  of  p substituted  in  (14)  will  give  an  equation  in- 
volving only  u,  the  solution  of  which  will  determine  Prob.  z, 
since  by  (13)  Prob.  z = u.  It  remains  to  assign  the  limits  of  u. 

11.  Now  the  very  same  analysis  by  which  the  limitswere  deter- 
mined in  the  particular  case  in  which  n - 2,  (XIX.  12)  con- 
ducts us  in  the  present  case  to  the  following  result.  The  quan- 
tity u,  in  order  that  it  may  represent  the  value  of  Prob.  z,  must 
must  have  for  its  inferior  limits  the  quantities  ax,  a2,  . . an,  and 
for  its  superior  limits  the  quantities  blt  b2, . . bn,  ax  + a2  . . + an. 
W e may  hence  infer,  a priori , that  there  will  always  exist  one 
root,  and  only  one  root,  of  the  equation  (14)  satisfying  these 
conditions.  I deem  it  sufficient,  for  practical  verification,  to  show 
that  there  will  exist  one,  and  only  one,  root  of  the  equation  (14), 
between  the  limits  ax,  a2,  . . an,  and  bx,  b2,  . .bn. 

First,  let  us  consider  the  nature  of  the  changes  to  which  p is 
subject  in  (15),  as  u varies  from  ax,  which  we  will  suppose  the 
greatest  of  its  minor  limits,  to  bi}  which  we  will  suppose  the  least 
of  its  major  limits.  When  u = a1}  it  is  evident  that  p is  positive 
and  greater  than  ax . When  u = bx,  we  have  p -bx,  which  is  also 
positive.  Between  the  limits  u = al,  u = bx,  it  may  be  shown 
that  p increases  with  u.  Thus  we  have 

dp  = j _ {bz  - u)  . . (bn  - u)  _ (Z>,  - u)  (b3  - u)  . . (bn  - u ) 
du  (l-w)"'1  (1  — u)n~l 

+ (n  _ n (fr  - u ) (k  -u)..(bn-  u ) (16) 

' ' (1  - u)n 

bx  - u bn  - u 

1 - u 1 - u 


Now  let 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


343 


Evidently  xlt  x2,  . . xn,  will  be  proper  fractions,  and  we  have 

~^j-~  — 1 X2X%  ■ . Xn  X\  X3  . . Xn  . . — X\  X2  • . Xn-\  — 1)  X\  X2  . . Xji 

= 1 - (1  - a;,)  x2  x3 . . xn  - xx  (1  - x2)  x3  . . xn  . . 

X\  X2  . . Xn_i  (1  — Xn)  — Xi  x2  . . xn. 

Now  the  negative  terms  in  the  second  member  are  (if  we  may 
borrow  the  language  of  the  logical  developments)  constituents 
formed  from  the  fractional  quantities  xlf  x2,  . . xn.  Their  sum 

ft 

cannot  therefore  exceed  unity ; whence  -J-  is  positive,  and  g in- 
creases with  u between  the  limits  specified. 

Now  let  (14)  be  written  in  the  form 

(n  - a,)  . . (g  - an) 


- (g-  u)  = 0, 


(17) 


(18) 


and  assume  u = av.  The  first  member  becomes 

and  this  expression  is  negative  in  value.  For,  making  the  same 
assumption  in  (15),  we  find 

( bx  - u)  . . (bn  - u ) 


M Ul  (1  - u)n-1 

At  the  same  time  we  have 


= a positive  quantity. 


(g  - a2)  . . (g  - a„)  g-a2  g-a.n 

n - 1 * * * 

M M M- 

and  since  the  factors  of  the  second  member  are  positive  fractions, 
that  member  is  less  than  unity,  whence  (18)  is  negative.  Where- 
fore the  assumption  u = al  makes  the  first  member  of  (17)  ne- 
gative. 

Secondly,  let  u - bx , then  by  (15)  g = u = bl}  and  the first  mem- 
ber of  (17)  becomes  positive. 

Lastly,  between  the  limits  u-ax  and  u = bi,  the  first  member 
of  (17)  continuously  increases.  For  the  first  term  of  that  ex- 
pression written  under  the  form 

ju  - a,  g-  an 


(M  ~ «i) 


t1 


344 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


increases,  since  p increases,  and,  with  it,  every  factor  contained. 
Again,  the  negative  term  p - u diminishes  with  the  increase  of 
u,  as  appears  from  its  value  deduced  from  (15),  viz., 

(6,  - u)  . . (bn-  u ) 

(1  - u)n~l 

Hence  then,  between  the  limits  u = u = bx,  the  first  member 
of  ( 1 7 ) continuously  increases,  changing  in  so  doing  from  a nega- 
tive to  a positive  value.  Wherefore,  between  the  limits  assigned, 
there  exists  one  value  of  u,  and  only  one,  by  which  the  said 
equation  is  satisfied. 

12.  Collecting  these  results  together,  we  arrive  at  the  follow- 
ing solution  of  the  general  problem. 

The  probability  of  the  event  E will  be  that  value  of  u de- 
duced from  the  equation 


wherein 

p = u + 


(jU  - ClPl)  . . (fJL  - cnpn ) 

3^7 — » 

r 

{ 1 - Cl  (1  - px)  - u)  . . { 1 - cn  (1  - pn)  - u) 
~~  (1  - u)n~l  ’ 


(19) 


which  (value)  lies  between  the  two  sets  of  quantities, 

cyp i,  c2p2,  . ,cnp„  and  1-^(1  -px),  1 - c2  (1  -p2)  . . l-c„(l -/>„), 

the  former  set  being  its  inferior,  the  latter  its  superior,  limits. 

And  it  may  further  be  inferred  in  the  general  case,  as  it  has 
been  proved  in  the  particular  case  of  n = 2,  that  the  value  of  u, 
determined  as  above,  will  not  exceed  the  quantity 

CiPi  + c2p2  . . + cnpn. 


13.  Particular  verifications  are  subjoined. 

1st.  Let  px  = 1,  p2  = 1,  . . pn  =>  1.  This  is  to  suppose  it  cer- 
tain, that  if  any  one  of  the  events  A2  . . A n,  happen,  the 
event  E will  happen.  In  this  case,  then,  the  probability  of  the 
occurrence  of  E will  simply  be  the  probability  that  the  events  or 
causes  Alf  A 2 . . An  do  not  all  fail  of  occurring,  and  its  expression 
will  therefore  be  1 - (1  - c,)  (1  - c2)  . . (1  - c„). 

Now  the  general  solution  (19)  gives 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


345 


- u = 


(p  - C0  • * (P  - °n) 


,n- 1 


wherein 

Hence, 


(1  - «)n  . 


1 - M = (1-  Cx)  . . (1  - C„), 

U = 1 - (1  - Cj)  . . (1  - Cn), 

equivalent  to  the  a 'priori  determination  above. 

2nd.  Letpj  =0,  p2  = 0,  pn  = 0,  then  (19)  gives 

p - U = p, 

.*.  u = 0, 

as  it  evidently  ought  to  be. 

3rd.  Let  c1}  c.2.  .cn  be  small  quantities,  so  that  their  squares 
and  products  may  be  neglected.  Then  developing  the  second 
members  of  the  equation  (19), 

pn  - (cipi  + c2p2  . . + c„pn ) pn-1 


p - u =- 


— p - (CiPi  + C2P2  • • + cnpn), 
.*.  u = cxpx  + c2p2  . . + cnpn. 


Now  this  is  what  the  solution  would  be  were  the  causes 
A1,  A2  . . An  mutually  exclusive.  But  the  smaller  the  proba- 
bilities of  those  causes,  the  more  do  they  approach  the  condition 
of  being  mutually  exclusive,  since  the  smaller  is  the  probability  of 
any  concurrence  among  them.  Hence  the  result  above  obtained 
will  undoubtedly  be  the  limiting  form  of  the  expression  for  the 
probability  of  E. 

4th.  In  the  particular  case  of  n = 2,  we  may  readily  elimi- 
nate p from  the  general  solution.  The  result  is 

(u  - c.p,)  ( u - c2p2 ) _ {1  - cx  (1  -pQ  - u)  {1  - c2  (1  - p2)  -u] 
CiPl  + c2p2  - u 1 -u 

which  agrees  with  the  particular  solution  before  obtained  for  this 
case,  Problem  1. 

Though  by  the  system  (19),  the  solution  is  in  general  made 
to  depend  upon  the  solution  of  an  equation  of  a high  order,  its 


346 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 

practical  difficulty  will  not  be  great.  For  the  conditions  relating 
to  the  limits  enable  us  to  select  at  once  a near  value  of  u,  and 
the  forms  of  the  system  (19)  are  suitable  for  the  processes  of  suc- 
cessive approximation. 

14.  Problem  7. — The  data  being  the  same  as  in  the  last  pro- 
blem, required  the  probability,  that  if  any  definite  and  given 
combination  of  the  causes  ^41?  A2,  . . An,  present  itself,  the  event 
E will  be  realized. 

The  cases  Ax,  A2,  . . An,  being  represented  as  before  by 
a?i,  x2,  . . xn  respectively,  let  the  definite  combination  of  them, 
referred  to  in  the  statement  of  the  problem,  be  represented  by 
the  <p  (#!,  x2  . ■ x„)  so  that  the  actual  occurrence  of  that  combi- 
nation will  be  expressed  by  the  logical  equation, 

('El  5 *2,  . . #7t)  = 1. 

The  data  are 

Prob  Xi  = Ci,  . . Prob. xn  = cn, 

Prob.  xxz  = Cipx,  Prob.  xnz  = cnpn  ; 
and  the  object  of  investigation  is 

Prob.  <j)  (#i,  x2 . . xn)  z 
Prob.  0 (xx,x2 . . xn) 

We  shall  first  seek  the  value  of  the  numerator. 

Let  us  assume, 

X\  Z t \ • • Xn  Z>  = ^715 
(j)  (^1  j X2  • . xA)  Z — W. 

Or,  if  for  simplicity,  we  represent  $ (xx,  x2  . . xn)  by  0,  the  last 
equation  will  be 

(pz  = w,  (5) 

to  which  must  be  added  the  equation 

X\  x<t  • • Xn  z = 0,  (6) 

Now  any  equation  xrz  = tr  of  the  system  (3)  may  be  reduced 
to  the  form 

XrZtr  + tr  (1  - XrZ)  = 0. 

Similarly  reducing  (5),  and  adding  the  different  results  together, 
we  obtain  the  logical  equation 


(1) 

(2) 

(3) 

(4) 


PROBLEMS  ON  CAUSES. 


CHAP.  XX.] 


347 


2 { XrZtr  + tr  (1  - Xrz)  } 4-  . . XnZ  4-  $ZW  + W (1  - <f>Z ) = 0,  (7) 

from  which  z being  eliminated,  w must  be  determined  as  a de- 
veloped logical  function  of  xx , . . xn , tx , . . t„  . 

Now  making  successively  2=1,  2 = 0 in  the  above  equation, 
and  multiplying  the  results  together,  we  have 


( 2 (xTtr  + xTtr ) + xx . . xn  + <pw  + w<p } x (27-  + w)  = 0. 


Developing  this  equation  with  reference  to  w,  and  replacing 
in  the  result  27-  + 1 by  l,  in  accordance  with  Prop.  i.  Chap,  ix., 
we  have 

Eio  + E'  (1  - w)  = 0 ; 

wherein 

Ej  = 2 ('Xytf  "i"  tfXr^  "T  Xl  . . Xjl  "(“  0, 


And  hence 


Ei  — 2 tf  f 2 (i Xj-tf  “f*  tj-Xf^  4“  X\  . . Xn  4"  (j) } . 


w = 


E 

E'-E ‘ 


(8) 


The  second  member  of  this  equation  we  must  now  develop 
with  respect  to  the  double  series  of  symbols  xx,  x2, . . xn,  tlf  t2, . . tn. 
In  effecting  this  object,  it  will  be  most  convenient  to  arrange 
the  constituents  of  the  resulting  development  in  three  distinct 
classes,  and  to  determine  the  coefficients  proper  to  those  classes 
separately. 

First,  let  us  consider  those  constituents  of  which  . . Tn  is  a 
factor.  Making  ti  = 0 . . tn  = 0,  we  find 


E’  = 0,  E - 2 Xr  4-  Xi  . . Xn  4-  0 . 


It  is  evident,  that  whatever  values  (0,  1)  are  given  to  the  «-sym- 
bols,  E does  not  vanish.  Hence  the  coefficients  of  all  constituents 
involving  Jx  . .Jn  are  0. 

Consider  secondly,  those  constituents  which  do  not  involve  the 
factor  7 . .Tn,  and  which  are  symmetrical  with  reference  to  the  two 
sets  of  symbols  xx . . xn  and  tx.  .tn.  By  symmetrical  constituents 
is  here  meant  those  which  would  remain  unchanged  if  xx  were 
converted  into  tx,  x2  into  t2,  &c.,  and  vice  versa.  The  constitu- 
ents xx  . . xn  tx  . . tn,  xx  . . xn  Ti  . . 7„,  &c.,  are  in  this  sense  sym- 
metrical. 


348 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


F or  all  symmetrical  constituents  it  is  evident  that 
2 (Xr  tr  tT 

vanishes.  For  those  which  do  not  involve  7,  . . 7„ , it  is  further 
evident  that  7,  . . x„  also  vanishes,  whence 

E=f,  E'  = 2tr(tf>), 

2tr(<t>) 

w = v 1 y__-  . 

2itr(<f>)  — <J> 


For  those  constituents  of  which  the  ^-factor  is  found  in  <p  the 
second  member  of  the  above  equation  becomes  1 ; for  those  of 
which  the  £ -factor  is  found  in  ip  it  becomes  0.  Hence  the  coeffi- 
cients of  symmetrical  constituents  not  involving  7,  . . 7„,  of  which 
the  x-f actor  is  found  in  (j>  ivill  be  1 ; of  those  of  which  the  x-factor 
is  not  found  in  ip  it  will  be  0. 

Consider  lastly,  those  constituents  which  are  unsymmetrical 
with  reference  to  the  two  sets  of  symbols,  and  which  at  the  same 
time  do  not  involve  7j  . . 7„ . 

Here  it  is  evident,  that  neither  E nor  E'  can  vanish,  whence 
the  numerator  of  the  fractional  value  of  w in  (8)  must  exceed 
the  denominator.  That  value  cannot  therefore  be  represented 


by  1,  0,  or 


0 

O’ 


It  must  then,  in  the  logical  development,  be  re- 


presented by  - . 


Such  then  will  be  the  coefficient  of  this  class 


of  constituents. 

15.  Hence  the  final  logical  equation  by  which  w is  expressed 
as  a developed  logical  function  of  xx,  . . xn,  tx,  . . tn,  will  be  of 
the  form 


w=  2,  (XT)  + 0 {S2(X77)  + tf . . tn]  (sum  of  other  con-  . 

^ stituents),  v*/ 

wherein  (X T)  represents  the  sum  of  all  symmetrical  consti- 
tuents of  which  the  factor  X is  found  in  0,  and  22  (X  T),  the 
sum  of  all  symmetrical  constituents  of  which  the  factor  X is  not 
found  in  <p, — the  constituent  xt . .xn  7,  . . 7„,  should  it  appear, 
being  in  either  case  rejected. 

Passing  from  Logic  to  Algebra,  it  may  be  observed,  that 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


349 


here  and  in  all  similar  instances,  the  function  V,  by  the  aid  of 
which  the  algebraic  system  of  equations  for  the  determination  of 
the  values  of  xif  . . xn,  tl}  . . t„  is  formed,  is  independent  of  the 
nature  of  any  function  <p  involved,  not  in  the  expression  of  the 
data , but  in  that  of  the  quasitum  of  the  problem  proposed.  Thus 
we  have  in  the  present  example, 


Prob.  w 


2,  (XT) 
V ’ 


wherein  V = 2|  (X  T)  + S2  ( X T)  + f,  . . tn 

= 2 (XT)  + 7i . . f„.  (10) 

Here  2 (XT)  represents  the  sum  of  all  symmetrical  constituents 
of  the  x and  t symbols,  except  the  constituent  ~x{  . . xn,  7,  . . 7„. 
This  value  of  V is  the  same  as  that  virtually  employed  in  the  so- 
lution of  the  preceding  problem,  and  hence  we  may  avail  our- 
selves of  the  results  there  obtained . 

If  then,  as  in  the  solution  referred  to,  we  assume 


X\t\  Xntn  Xi  „ 

=-=  = »*,,  =-=■  = rn„,  — = &c., 

X\  Xji  tji  X\ 

we  shall  obtain  a result  which  may  be  thus  written  : 

Mi 


Prob. w = 


M+N-  1’ 


(H) 


Mi  being  formed  by  rejecting  from  the  function  0 the  constituent 
Xi . . xn,  if  it  is  there  found,  dividing  the  result  by  the  same  con- 

stituent  Xi . . xn,  and  then  changing  ~ into  to,,  — into  m2,  and 

X\  X2 

so  on.  The  values  of  M and  N are  the  same  as  in  the  preceding 
problem.  Reverting  to  these  and  to  the  corresponding  values  of 
ml}  m2,  &c.,  we  find 

Prob  . w = Mi(fj  + v-  1), 


the  general  values  of  mr,  nr  being 


mr 


crpr 

im  - trpT 


Cr(l  -Pr) 

H - Cr  (1  - Pr)’ 


and  fi  and  v being  given  by  the  solution  of’  the  system  of  equa- 
tions, 


350  PROBLEMS  ON  CAUSES.  [CHAP.  XX. 

nil  l-(t*-CiPi)"(f*-CnPn)  { V~Cj  ( 1 -joQ  j . . { V- C„(  1 - pn) } 

/lT1  v"-1 

The  above  value  of  Prob.  w will  be  the  numerator  of  the  fraction 
(2).  It  now  remains  to  determine  its  denominator. 

For  this  purpose  assume 

<P  (®i,  X2  . . X n)  = U, 


or  <p  = v ; 

whence  <pv  + v$  =0. 

Substituting  the  first  member  of  this  equation  in  (7 ) in  place  of 
the  corresponding  form  <pzw  + w (1  - <f>z)  we  obtain  as  the  primary 
logical  equation, 

2 {xr  z lr  + tr  ( 1 - xr  z) } + xi . . xnz  + <pl  + vlp  - 0, 
whence  eliminating  z,  and  reducing  by  Prop.  n.  Chap.  IX., 

(f)V  + V^+  2£r  ( 2 ( Xr  lr  + tr  Xr)  + Xi  . . xn}  = 0. 

Hence 

<b  + 2£r  { S,(xr7r  + trXr)  + ~X\  . . Xn) 

V = 2<p  - 1 ’ 

and  developing  as  before, 


w = Si  (X  T)  + h . . Tn  2i/X)  + 0 { S2  ( X T)  + 1, . . 7n  S2  (X) ) 


+ - (sum  of  other  constituents). 


(12) 


Here  2i  ( X ) indicates  the  sum  of  all  constituents  found  in  <p, 
S2(X)  the  sum  of  all  constituents  not  found  in  (j>.  The  expres- 
sions are  indeed  used  in  place  of  $ and  1 - <j>  to  preserve  sym- 
metry. 

It  follows  hence  that  Si  (X)  + S2(X)  = 1,  and  that,  as  be- 
fore, Si  (X  T)  + S2  (X  T)  = S ( X T).  Hence  V will  have  the 
same  value  as  before,  and  we  shall  have 


Prob.  v = 


2,  (XT)  +7i..f„  Si  (X) 
V 


Or  transforming,  as  in  the  previous  case, 

+ If i 


Prob.  v = 


M+N-  1’ 


(13) 


CHAP.  XX.]  PROBLEMS  ON  CAUSES. 


351 


wherein  Nr  is  formed  by  dividing  <p  by  xx . . xn,  and  changing  in 
oc  oc 

the  result  J-  into  n,.  =?  into  n 2,  &c. 

Xi  x2 

Now  the  final  solution  of  the  problem  proposed  will  be  given 
by  assigning  their  determined  values  to  the  terms  of  the  fraction 

Prob.  6 (Xi , . . xn)  z Prob.  w 

1 1 Qp  # 

Prob.  $ (xx , . . x„y  Prob.  v ' 

Hence,  therefore,  by  (11)  and  (13)  we  have 

Prob.  sought  = r— . 

6 Mi  + Ni 


A very  slight  attention  to  the  mode  of  formation  of  the  func- 
tions Mi  and  Ni  will  show  that  the  process  may  be  greatly  sim- 
plified. We  may,  indeed,  exhibit  the  solution  of  the  general 
problem  in  the  form  of  a rule,  as  follows  : 


Reject  from  the  function  (p  (arls  x2 . . x„)  the  constituent  xx . .xn  if 
it  is  therein  contained , suppress  in  all  the  remaining  constituents 
the  factors  xXJ  x2,  §*c.,  and  change  generally  in  the  result  xT  into 

— — — — . Call  this  result  Mx. 
fl  — Cr  Pr 

Again , replace  in  the  function  <p  (xx,  x2 . . xn)  the  constituent 
xx . . xn  if  it  is  therein  found , by  unity ; suppress  in  all  the  remaining 
constituents  the  factors  xx , x2,  fyc.,  and  change  generally  in  the  re- 
sult xr  into  ■ °r  ^ 

V — Cr(l  -pr) 

Then  the  solution  required  will  be  expressed  by  the  formula 

Mi 

Mi  + Nj 


g and  v being  determined  by  the  solution  of  the  system  of  equations 


I 1 ••  (n~cnPn) 

/i”'1 

= \ V-Cn{\-Pn))  ^ 


It  may  be  added,  that  the  limits  of  y and  v are  the  same  as  in 
the  previous  problem.  This  might  be  inferred  from  the  general 
principle  of  continuity;  but  conditions  of  limitation,  which  are 


PROBLEMS  ON  CAUSES. 


352 


[chap.  XX. 


probably  sufficient,  may  also  be  established  by  other  conside- 
rations. 

Thus  from  the  demonstration  of  the  general  method  in  pro- 
babilities, Chap.  XVII.  Prop,  iv.,  it  appears  that  the  quantities 
xx , . . xn,  tx,  . . tn,  in  the  primary  system  of  algebraic  equations, 
must  he  positive  proper  fractions.  Now 

xr  cr  (1  — pr ) 

1 = nr  = V 

1 Xp  V Cp  ^1  Pr) 

Hence  generally  nr  must  be  a positive  quantity,  and  therefore 
we  must  have 

vjcr(l  -p,). 

In  like  manner  since  we  have 

Xr  tip  Cp  Pp 

= JPI  , — z 

( 1 — arr)  [ 1 — ^r)  fi  — cr  pr 


we  must  have  generally 


fX  > Cr  Pr  • 


16.  It  is  probable  that  the  two  classes  of  conditions  thus  re- 
presented are  together  sufficient  to  determine  generally  which  of 
the  roots  of  the  equations  determining  fx  and  v are  to  be  taken. 
Let  us  take  in  particular  the  case  in  which  n = 2.  Here  we  have 

{fi-cxpi)  (n~c2p2)  cxpxc2p2 

ju  + v - 1 = — — — — = n - (cxpx  + c2p2)  + — 

fX  p 

C\  px  c2  p2  (n  — Ci  pi)  c2 p2 

.*.  V = 1 - Cl  Pi  - c2p2  + — — = 1 - CiPi  - — — — . 

q H 

Whence,  since  fi>CiPi  we  have  generally 

v < 1 - Cipi. 

In  like  manner  we  have 


v<l-c2p2,  n < l - Ci(l  - pi),  fi  < 1 - c2(l  -p2). 

Now  it  has  already  been  shown  that  there  will  exist  but  one 
value  of  fi  satisfying  the  whole  of  the  above  conditions  relative 
to  that  quantity,  viz. 

fX  > Cr  pry  fX  < 1 — Cr(l  ~ p i f 

whence  the  solution  for  this  case,  at  least,  is  determinate.  And  I 


PROBLEMS  ON  CAUSES. 


353 


CHAP.  XX.] 

apprehend  that  the  same  method  is  generally  applicable  and  suf- 
ficient. But  this  is  a question  upon  which  a further  degree  of 
light  is  desirable. 

To  verify  the  above  results,  suppose  <f>  (r15 . .xn)  = 1,  which  is 
virtually  the  case  considered  in  the  previous  problem.  Now  the 
development  of  1 gives  all  possible  constituents  of  the  symbols 
Xi, . . xn.  Proceeding  then  according  to  the  Rule,  we  find 

Mi  = r - 1 = - — - - 1 by  (15). 

(n  - Ci  Pi)  . . (,u  - cn  pn)  n + V - 1 J v ' 

— l _ v ] 

(v-CjO-JB,)}  ..{v-Cn(l-pn)}  ‘ fjL+V-  1 

Substituting  in  (14)  we  find 

Prob.  z = 1 - v, 

which  agrees  with  the  previous  solution. 

Again,  let  <f>  (x1} . . xn)  = which,  after  development  and  sup- 
pression of  the  factors  x2,  . . xn,  gives  x,  (x2  + 1)  . . (xn  +1),  whence 
we  find 


Mi  = 
Ni  = 


'Jluf , _*£!_  by  (15). 

(n-CipO  . . (n~cnpn)  fi  + V - 1 

Cj{l-pi)vn-' = Ci  (1  -p^ 

{v-C^l  -/?i))..  jv-C„(l  -pn))  p+V-l 


Substituting,  we  have 

Probability  that  if  the  event  A i occur,  E will  occur  = px. 


And  this  result  is  verified  by  the  data.  Similar  verifications 
might  easily  be  added. 

Let  us  examine  the  case  in  which 


< p (*Ti, . . a*n)  — X\  x2 . . xn  "I-  x2  X\  x2 . . Xji  . . Xn  X\ . . xn_\. 
Here  we  find 

HjT  ^1  P 1 Pn 

Mi  = . . h , 

p-CiPi  p-Cnpn 

N = IC‘Q  ~PQ  + c»(l  ~P*)  . 

V — Ci  (1  — pi ) V — c„  (1  — pn ) 

whence  we  have  the  following  result — 


354 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


Probability  that  if  some  one  • 
alone  of  the  causes  AuA2..An  1 
present  itself,  the  event  E f 
will  follow. 


2 CrPr 

H ~ Cr  p, 

2 cr'Pr  + ^ Cr(l  ~ fr) 

P — Crpr  V-Cr(l  ” Pr ) 


Let  it  be  observed  that  this  case  is  quite  different  from  the 
well-known  one  in  which  the  mutually  exclusive  character  of 
the  causes  Ax, . . An  is  one  of  the  elements  of  the  data,  expressing 
a condition  under  which  the  very  observations  by  which  the  pro- 
babilities of  Au  A2,  &c.  are  supposed  to  have  been  determined, 
were  made. 

Consider,  lastly,  the  case  in  which  <j>  (xlt  ..xn)  = x1x2..  xn. 
Here 

M Cj  pl . . cn  pn  = Ci  pi . . cn  pn 

1 (p  - Ci  pi)  . . (p  - Cn  pn)  pn'1(p  + v-  1)’ 

Ci  (1  - pi)  . . CB  (1  - pn)  = C'(l  ~ Px)  ■ -C»(l  ~pn) 
jv  - Ci  (1  -JOl)j  ..  {v-c„(l-p„)j  v"'1  {p  + v - 1) 


Hence  the  following  result — 

Probability  that  if  all  the  " 

causes  Ai,  . . A„  con-  p, . .pnvn~l 

spire,  the  event  E will  [ pi . . pnvn + (1  -pi)  . . (1  -pn)  a t*"1’ 
follow. 


This  expression  assumes,  as  it  ought  to  do,  the  value  1 when  any 
one  of  the  quantities  p\, . .pn  is  equal  to  1. 

17.  Problem  VIII. — Certain  causes  Ai,  A2..An  being  so 
restricted  that  they  cannot  all  fail,  but  still  can  only  occur  in  cer- 
tain definite  combinations  denoted  by  the  equation 

<p  (-d-i,  A2  • • An^  = 1, 

and  there  being  given  the  separate  probabilities  cx , . . c„  of  the 
said  causes,  and  the  corresponding  probabilities  px,  . . pn  that  an 
event  E will  follow  if  those  respective  causes  are  realized,  re- 
quired the  probability  of  the  event  E. 

This  problem  differs  from  the  one  last  considered  in  several 
particulars,  but  chiefly  in  this,  that  the  restriction  denoted  by  the 
equation  <p  (^4l5 . . An)  = 1,  forms  one  of  the  data,  and  is  supposed 


PROBLEMS  ON  CAUSES. 


355 


CHAP.  XX.] 


to  be  furnished  by  or  to  be  accordant  with  the  very  experience 
from  which  the  knowledge  of  the  numerical  elements  of  the 
problem  is  derived. 

Representing  the  events  Alt . . An  by  xlt . . xn  respectively, 
and  the  event  JE  by  2,  we  have — 

Prob.  xT  = cr,  Prob.  xrz  = c,pT.  (1) 

Let  us  assume,  generally, 

OC'p  Z = 


then  combining  the  system  of  equations  thus  indicated  with  the 
equations 

xl . . xn  = 0,  # ( xx , . . x„)  = 1,  or  0 = 1, 

furnished  in  the  data,  we  ultimately  find,  as  the  developed  ex- 
pression of  z, 

z=2(XT)  + 0ti72..7„Z(X),  (2) 


where  X represents  in  succession  each  constituent  found  in  <p, 
and  T a similar  series  of  constituents  of  the  symbols  tu . . tn ; 
2 (XT)  including  only  symmetrical  constituents  with  reference 
to  the  two  sets  of  symbols. 

The  method- of  reduction  to  be  employed  in  the  present  case 
is  so  similar  to  the  one  already  exemplified  in  former  problems, 
that  I shall  merely  exhibit  the  results  to  which  it  leads.  We 
find 


Prob.  2 


M 

M + N ’ 


(3) 


with  the  relations 


Mi 

C\fl 


Mn_  = = 

CnPn  Ci(l  ~ P\)  C«(l  — Pn) 


M+N. 


(4) 


Wherein  M is  formed  by  suppressing  in  <p  (xu  . . xn)  all  the  fac- 
tors xi, . .xn,  and  changing  in  the  result  X\  into  m\,  xn  into  mn, 
while  N is  formed  by  substituting  in  M,  n\  for  m\ , &c. ; more- 
over Mi  consists  of  that  portion  of  M of  which  mi  is  a factor, 
Ni  of  that  portion  of  N of  which  ri\  is  a factor ; and  so  on. 

Let  us  take,  in  illustration,  the  particular  case  in  which  the 
causes  Ax . . An  are  mutually  exclusive.  Here  we  have 

(p  (Xl,  . . Xn)  = Xx  X2  . . Xn  . . . + x„  Xl  . . Xn.x. 


356 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


Whence 

M = my  + m2 . . + mn, 
N = nx  + n2  . . + n„ , 
Mx-  mx,  Ny  = riy,  &c. 
Substituting,  we  have 


m\  _mn  ny  nn 

C\P\  ' ’ cnpn  c,(l  -pi)  c„(l  -pn) 

Hence  we  find 


my  + m2  . . + mn 


C\  P\  + c,  p2  . . + cn  pn 


= M+  N, 


M+  N. 


or 

Hence,  by  (3), 


M 


C\p i • • + cnpn 


M+  N. 


Prob.  Z = Cypy  . . 4 cnp„, 


a known  result. 

There  are  other  particular  cases  in  which  the  system  (4)  ad- 
mits of  ready  solution.  It  is,  however,  obvious  that  in  most 
instances  it  would  lead  to  results  of  great  complexity.  Nor  does 
it  seem  probable  that  the  existence  of  a functional  relation  among 
causes,  such  as  is  assumed  in  the  data  of  the  general  problem,  will 
often  be  presented  in  actual  experience ; if  we  except  only  the 
particular  cases  above  discussed. 

Had  the  general  problem  been  modified  by  the  restriction 
that  the  event  E cannot  occur,  all  the  causes  Ay. . An  being  ab- 
sent, instead  of  the  restriction  that  the  said  causes  cannot  all  fail, 
the  remaining  condition  denoted  by  the  equation  0(^1 1,  . . A n)  = 1 
being  retained,  we  should  have  found  for  the  final  logical  equation 


2=2,  (XT)  + 0 2(X), 

2 (X)  being,  as  before,  equal  to  $ (#„  . . xn),  but  2,  (X T ) formed 
by  rejecting  from  (j>  the  particular  constituent  5, . . xn  if  therein 
contained,  and  then  multiplying  each  ^-constituent  of  the  result 
by  the  corresponding  ^-constituent.  It  is  obvious  that  in  the  par- 
ticular case  in  which  the  causes  are  mutually  exclusive  the  value 
of  Prob.  z hence  deduced  will  be  the  same  as  before. 


18.  Problem  IX. — Assuming  the  data  of  any  of  the  pre- 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


357 


vious  problems,  let  it  be  required  to  determine  the  probability 
that  if  the  event  E present  itself,  it  will  be  associated  with  the 
particular  cause  A,.;  in  other  words,  to  determine  the  a posteriori 
probability  of  the  cause  Ar  when  the  event  E has  been  observed 
to  occur. 

In  this  case  we  must  seek  the  value  of  the  fraction 


Prob.  xT  z 
Prob.  z ’ 


or 


Cr  pr 

Prob.  z’ 


by  the  data. 


(0 


As  in  the  previous  problems,  the  value  of  Prob.  z has  been  as- 
signed upon  different  hypotheses  relative  to  the  connexion  or 
want  of  connexion  of  the  causes,  it  is  evident  that  in  all  those 
cases  the  present  problem  is  susceptible  of  a determinate  solution 
by  simply  substituting  in  (1)  the  value  of  that  element  thus  de- 
termined. 

If  the  a priori  probabilities  of  the  causes  are  equal,  we  have 
C[  = c2 . . = cr.  Hence  for  the  different  causes  the  value  (1)  will 
vary  directly  as  the  quantity^?,..  Wherefore  whatever  the  nature 
of  the  connexion  among  the  causes,  the  a posteriori  probability  of 
each  cause  will  be  proportional  to  the  probability  of  the  observed 
event  E when  that  cause  is  known  to  exist.  The  particular  case 
of  this  theorem,  which  presents  itself  when  the  causes  are  mu- 
tually exclusive,  is  well  known.  We  have  then 

Prob.  xrz  _ crpr  pr 

Prob.  Z 2<?rPr  Pl  + Pi  •■  + Pn 

the  values  of  c15  . . cn  being  equal. 

Although,  for  the  demonstration  of  these  and  similar  theo- 
rems in  the  particular  case  in  w'hich  the  causes  are  mutually  ex- 
clusive, it  is  not  necessary  to  introduce  the  functional  symbol  0, 
which  is,  indeed,  to  claim  for  ourselves  the  choice  of  all  possible 
and  conceivable  hypotheses  of  the  connexion  of  the  causes,  yet, 
under  every  form,  the  solution  by  the  method  of  this  work  of 
problems,  in  which  the  number  of  the  data  is  indefinitely  great, 
must  always  partake  of  a somewhat  complex  character.  Whe- 
ther the  systematic  evolution  which  it  presents,  first,  of  the  logi- 
cal, secondly,  of  the  numerical  relations  of  a problem,  furnishes 
any  compensation  for  the  length  and  occasional  tediousness  of  its 


358 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 

processes,  I do  not  presume  to  inquire.  Its  chief  value  undoubt- 
edly consists  in  its  power, — in  the  mastery  which  it  gives  us  over 
questions  which  would  apparently  baffle  the  unassisted  strength 
of  human  reason.  For  this  cause  it  has  not  been  deemed  super- 
fluous to  exhibit  in  this  chapter  its  application  to  problems,  some 
of  which  may  possibly  be  regarded  as  repulsive,  from  their  diffi- 
culty. without  being  recommended  by  any  prospect  of  immediate 
utility.  Of  the  ulterior  value  of  such  speculations  it  is,  I con- 
ceive, impossible  for  us,  at  present,  to  form  any  decided  judg- 
ment. 

19.  The  following  problem  is  of  a much  easier  description 
than  the  previous  ones. 

Problem  X. — The  probability  of  the  occurrence  of  a certain 
natural  phenomenon  under  given  circumstances  is  p.  Observation 
has  also  recorded  a probability  a of  the  existence  of  a permanent 
cause  of  that  phenomenon,  i.e.  of  a cause  ivhich  would  always  pro- 
duce the  event  under  the  circumstances  supposed.  What  is  the 
probability  that  if  the  phenomenon  is  observed  to  occur  n times  in 
succession  under  the  given  circumstances , it  will  occur  the  n + ltA 
time  ? What  also  is  the  probability,  after  such  observation , of  the 
existence  of  the  permanent  cause  referred  to  ? 

First  Case. — Let  t represent  the  existence  of  a permanent 
cause,  and  xx,  x2  . . xn+l  the  successive  occurrences  of  the  natural 
phaenomenon. 

If  the  permanent  cause  exist,  the  events  xx , x2 . . xn+l  are  ne- 
cessary consequences.  Hence 

t = vxx,  t = vx2,&  c., 

and  eliminating  the  indefinite  symbols, 

#(l-a;1)  = 0,  <(l-r2)  = 0,  t (1  - x^x)  = 0. 

Now  we  are  to  seek  the  probability  that  if  the  combination 
i)!,..  xn  happen,  the  event  x„+x  will  happen,  i.  e.  we  are  to  seek 
the  value  of  the  fraction 

Prob.  *i  I,  . . &Vi 
Prob.  xx  x2 . . xn 


We  will  first  seek  the  value  of  Prob.  xxxz. . xn, 


PROBLEMS  ON  CAUSES. 


359 


CHAP.  XX.] 


Represent  the  combination  x2 . .xn  by  w,  then  we  have  the 
following  logical  equations : 


£(1  - xx)  = 0,  *(1  - x2)  = 0 . . t (1  - x„ ) = 0, 

Xi  X2  . . Xn  = w. 

Reducing  the  last  to  the  form 


(a?!  Xi  . . Xn)  (1  - w)  + V)  (1  - X2  . . Xn)  = 0, 

and  adding  it  to  the  former  ones,  we  have 

2£(1  - Xi)  + Xj  X2  . . Xn  (1  - w)  + W (1  - Xi  x2..xn)  = 0,  (1) 

wherein  2 extends  to  all  values  of  i from  1 to  n,  for  the  one  logi- 
cal equation  of  the  data.  With  this  we  must  connect  the  nume- 
rical conditions, 

Prob.  X\  = Prob.  x2 . . = Prob.  x„  = p,  Prob.  t = a ; 
and  our  object  is  to  find  Prob.  w. 

Prom  (1)  we  have 

2 1 (1  - Xi)  + xx  x2 . . xn 

w = ^ z ; 

2 Xi  X2  . . Xn  - 1 


2 (1  - Xi)  + x1x2..xn 
2 xx  x2 . . x„  - 1 


t + 


XX  X2  • • X\ 

2 xx  x2 . . xn 


(2) 


on  developing  with  respect  to  t.  This  result  must  further  be 
developed  with  respect  to  x1}  x2, . . xn. 

Now  if  we  make  ajx  = 1,  x2  = 1,  . . xn  = 1,  the  coefficients  both 
of  t and  of  1 - t become  1.  If  we  give  to  the  same  symbols  any 
other  set  of  values  formed  by  the  interchange  of  0 and  1 , it  is 
evident  that  the  coefficient  of  t will  become  negative,  while  that 
of  1 - t will  become  0.  Hence  the  full  development  (2)  will  be 


w - Xi  x2 . . xnt  + xx  x2 . . xn  (1  - t)  + 0 (1  - Xi  x2 . . xn)  (1  - t) 

+ constituents  whose  coefficients  are  or  equivalent  to 
Here  we  have 

V = X!  x2  . . xnt  + Xx  X2  . . Xn  (1  - t)  + (1  - Xi  x2 . . xn)  (1  - t) 

= Xx  x2 . . xnt  + 1 - t ; 


whence,  passing  from  Logic  to  Algebra, 


360 


PROBLEMS  ON  CAUSES. 


[chap.  XX. 


Xj  X2  . . Xnt  4-  3?,(1  - t)  _Xx  X2  ..  Xnt  + X2  (1  - t ) 

P ~ P 

X\  X2  . . Xnt  -j-  Xji  [1  t\  X\  X2  . . Xnt  _ 

= L = — = Xx  X2  . . Xnt  + 1 

p a 


t. 


Prob.  w = 


XjX-z  . . Xn 
Xx  X2  . . X nt  + 1 - t 


From  the  forms  of  the  above  equations  it  is  evident  that  we 
have  Xy  = x2  . . = xn.  Replace  then  each  of  these  quantities  by  x, 
and  the  system  becomes 

xnt  + (l-t)x  x”t  , 

= — = xnt  + 1 - t, 

p a 

~jn 

Prob.  w = 


xnt  + 1 - t 

from  which  we  readily  deduce 

Prob.  w = Prob.  xx  x2 . . x„  = a + (p  - a)  ^ y — -J 
If  in  this  result  we  change  n into  n + 1,  we  get 

Prob.  xyx2..  xn+x  = a + (p  - a)  ^ y-^ 
Hence  we  find — 

Prob.  XxX2  . . xn+1  _a  + ^ ~ a^(l  -g) 

Prob.  xxx2. . x„ 


\n-i 


(3) 


as  the  expression  of  the  probability  that  if  the  phsenomenon  be  n 
times  repeated,  it  will  also  present  itself  the  n + Xth  time.  By  the 
method  of  Chapter  XIX.  it  is  found  that  a cannot  exceed  p in 
value. 

The  following  verifications  are  obvious  : — 

1st.  If  a = 0,  the  expression  reduces  to  p,  as  it  ought  to  do. 
For  when  it  is  certain  that  no  permanent  cause  exists,  the  suc- 
cessive occurrences  of  the  phenomenon  are  independent. 

2nd.  If  p=  1,  the  expression  becomes  1,  as  it  ought  to  do. 

3rd.  If  p = a,  the  expression  becomes  1,  unless  a = 0.  If  the 
probability  of  a phenomenon  is  equal  to  the  probability  that  there 


PROBLEMS  ON  CAUSES. 


361 


CHAP.  XX.] 

exists  a cause  which  under  given  circumstances  would  always 
produce  it,  then  the  fact  that  that  phenomenon  has  ever  been  no- 
ticed under  those  circumstances,  renders  certain  its  re-appearance 
under  the  same.* 

4th.  As  n increases,  the  expression  approaches  in  value  to 
unity.  This  indicates  that  the  probability  of  the  recurrence  of 
the  event  increases  with  the  frequency  of  its  successive  appear- 
ances,— a result  agreeable  to  the  natural  laws  of  expectation. 

Second  Case. — We  are  now  to  seek  the  probability  a.  pos- 
teriori of  the  existence  of  a permanent  cause  of  the  phenomenon. 
This  requires  that  we  ascertain  the  value  of  the  fraction 

Prob.ta]  x*. . xn 
Prob.  Xj  x2 . . x^ 

the  denominator  of  which  has  already  been  determined. 

To  determine  the  numerator  assume 

txx  x2 . . xn  = w, 

then  proceeding  as  before,  we  obtain  for  the  logical  develop- 
ment, 

w = tXi  x2 . . xn  + 0 (1  - t). 

Whence,  passing  from  Logic  to  Algebra,  we  have  at  once 

Prob.  w = a, 

a result  which  might  have  been  anticipated.  Substituting  then 
for  the  numerator  and  denominator  of  the  above  fraction  their 
values,  we  have  for  the  a posteriori  probability  of  a permanent 
cause,  the  expression 


* As  we  can  neither  re-enter  nor  recall  the  state  of  infancy,  we  are  unable  to 
say  how  far  such  results  as  the  above  serve  to  explain  the  confidence  with  which 
young  children  connect  events  whose  association  they  have  once  perceived. 
But  we  may  conjecture,  generally,  that  the  strength  of  their  expectations  is 
due  to  the  necessity  of  inferring  (as  a part  of  their  rational  nature),  and  the 
narrow  hut  impressive  experience  upon  which  the  faculty  is  exercised.  Hence 
the  reference  of  every  kind  of  sequence  to  that  of  cause  and  effect.  A little 
friend  of  the  author’s,  on  being  put  to  bed,  was  heard  to  ask  his  brother  the 
pertinent  question, — “ Why  does  going  to  sleep  at  night  make  it  light  in  the 
morning?”  The  brother,  who  was  a year  older,  was  able  to  reply,  that  it 
would  be  light  in  the  morning  even  if  little  boys  did  not  go  to  sleep  at  night. 


362 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


a 


a + (p  - a) 


p - a 
l - a 


It  is  obvious  that  the  value  of  this  expression  increases  with  the 
value  of  n. 

I am  indebted  to  a learned  correspondent,*  whose  original 
contributions  to  the  theory  of  probabilities  have  already  been  re- 
ferred to,  for  the  following  verification  of  the  first  of  the  above 
results  (3). 

“ The  whole  a priori  probability  of  the  event  (under  the  cir- 
cumstances) being  p , and  the  probability  of  some  cause  C which 
would  necessarily  produce  it,  a,  let  x be  the  probability  that  it 
will  happen  if  no  such  cause  as  C exist.  Then  we  have  the 
equation 

p - a + (1  - a)  x, 

whence  _ p-a 


Now  the  phenomenon  observed  is  the  occurrence  of  the  event  n 
times.  The  a priori  probability  of  this  would  be — 


1 supposing  C to  exist, 
xn  supposing  C not  to  exist ; 

whence  the  a posteriori  probability  that  C exists  is 


a 

a + (1  - a)*"’ 

that  C does  not  exist  is 

(1  - a)  xn 
a + (1  - a)  xn' 

Consequently  the  probability  of  another  occurrence  is 
a , ( 1 - a)  xn 

7- r X I + — r X a, 

a + (1  - a)  xn  a + (1  - a)  xn 


or 


a + (1  - a)  xnn 
a + (1  - a)  x11  ’ 


* Professor  Donkin. 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


363 


which,  on  replacing  n by  its  value 


p - a 
1 - a 


, will  be  found  to  agree 


with  (3).” 

Similar  verifications  might,  it  is  probable,  also  be  found  for 
the  following  results,  obtained  by  the  direct  application  of  the 
general  method. 

The  probability,  under  the  same  circumstances,  that  if,  out  of 
n occasions,  the  event  happen  r times,  and  fail  n - r times,  it  will 
happen  on  the  n + 1<A  time  is 


a + m (p  - a) 


a + m(p  - la) 


. . n(n-Y)..n-r+\  r 

wherein  m = -r—. and  l = 

1.2  . .r  n 


The  probability  of  a permanent  cause  (r  being  less  than  n) 
is  0.  This  is  easily  verified. 

If  p be  the  probability  of  an  event,  and  c the  probability  that 
if  it  occur  it  will  be  due  to  a permanent  cause ; the  probability 
after  n successive  observed  occurrences  that  it  will  recur  on  the 
n + 1th  similar  occasion  is 


wherein  x = 


~ c) 

1 - cp 


c + ( 1 - c)  xn 
c + (1  - c)  a:""1’ 


20.  It  is  remarkable  that  the  solutions  of  the  previous  pro- 
blems are  void  of  any  arbitrary  element.  W e should  scarcely, 
from  the  appearance  of  the  data,  have  anticipated  such  a circum- 
stance. It  is,  however,  to  be  observed,  that  in  all  those  problems 
the  probabilities  of  the  causes  involved  are  supposed  to  be  known 
a priori.  In  the  absence  of  this  assumed  element  of  knowledge, 
it  seems  probable  that  arbitrary  constants  would  necessarily  ap- 
pear in  the  final  solution.  Some  confirmation  of  this  remark  is 
afforded  by  a class  of  problems  to  which  considerable  attention 
has  been  directed,  and  which,  in  conclusion,  I shall  briefly 
consider. 


364  PROBLEMS  ON  CAUSES.  [CHAP.  XX. 

It  has  been  observed  that  there  exists  in  the  heavens  a large 
number  of  double  stars  of  extreme  closeness.  Either  these  ap- 
parent instances  of  connexion  have  some  physical  ground  or  they 
have  not.  If  they  have  not,  we  may  regard  the  phamomenon  of  a 
double  star  as  the  accidental  result  of  a “ random  distribution”  of 
stars  over  the  celestial  vault,  i.  e.  of  a distribution  which  would 
render  it  just  as  probable  that  either  member  of  the  binary  sys- 
tem should  appear  in  one  spot  as  in  another.  If  this  hypothesis  be 
assumed,  and  if  the  number  of  stars  of  a requisite  brightness  be 
known,  we  can  determine  what  is  the  probability  that  two  of 
them  should  be  found  within  such  limits  of  mutual  distance  as 
to  constitute  the  observed  phtenomenon.  Thus  Mitchell,*  esti- 
mating that  there  are  230  stars  in  the  heavens  equal  in  brightness 
to  j3  Capricorni,  determines  that  it  is  80  to  1 against  such  a 
combination  being  presented  were  those  stars  distributed  at  ran- 
dom. The  probability,  when  such  a combination  has  been  ob- 
served, that  there  exists  between  its  members  a physical  ground 
of  connexion,  is  then  required. 

Again,  the  sum  of  the  inclinations  of  the  orbits  of  the  ten 
known  planets  to  the  plane  of  the  ecliptic  in  the  year  1801  was 
9 1°‘4 187,  according  to  the  French  measures.  Were  all  inclina- 
tions equally  probable,  Laplacej  determines,  that  there  would  be 
only  the  excessively  small  probability  .00000011235  that  the 
mean  of  the  inclinations  should  fall  within  the  limit  thus  as- 
signed. And  he  hence  concludes,  that  there  is  a very  high 
probability  in  favour  of  a disposing  cause,  by  which  the  inclina- 
tions of  the  planetary  orbits  have  been  confined  within  such  narrow 
bounds.  Professor  De  Morgan,  J taking  the  sum  of  the  inclina- 
tions at  92°,  gives  to  the  above  probability  the  value  .00000012, 
and  infers  that  “ it  is  1 : .00000012,  that  there  was  a necessary 
cause  in  the  formation  of  the  solar  system  for  the  inclinations 
being  what  they  are.”  An  equally  determinate  conclusion  has 
been  drawn  from  observed  coincidences  between  the  direction  of 


* Phil.  Transactions,  An.  1767. 
f Theorie  Analytique  des  Probabilites,  p.  276. 
t Encyclopaedia  Metropolitana.  Art.  Probabilities. 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


365 


circular  polarization  in  rock-crystal,  and  that  of  certain  oblique 
faces  in  its  crystalline  structure.* 

These  problems  are  all  of  a similar  character.  A certain  hypo- 
thesis is  framed,  of  the  various  possible  consequences  of  which 
we  are  able  to  assign  the  probabilities  with  perfect  rigour.  Now 
some  actual  result  of  observation  being  found  among  those  con- 
sequences, and  its  hypothetical  probability  being  therefore  known, 
it  is  required  thence  to  determine  the  probability  of  the  hypo- 
thesis assumed,  or  its  contrary.  In  Mitchell’s  problem,  the  hy- 
pothesis is  that  of  a “ random  distribution  of  the  stars,” — the 
possible  and  observed  consequence,  the  appearance  of  a close 
double  star.  The  very  small  probability  of  such  a result  is  held 
to  imply  that  the  probability  of  the  hypothesis  is  equally  small, 
or,  at  least,  of  the  same  order  of  smallness.  And  hence  the  high 
and,  and  as  some  think,  determinate  probability  of  a disposing 
cause  in  the  stellar  arrangements  is  inferred.  Similar  remarks 
apply  to  the  other  examples  adduced. 

21.  The  general  problem,  in  whatsoever  form  it  may  be  pre- 
sented, admits  only  of  an  indefinite  solution.  Let  a:  represent  the 
proposed  hypothesis,  y a phtenomenon  which  might  occur  as  one 
of  its  possible  consequences,  and  whose  calculated  probability,  on 
the  assumption  of  the  truth  of  the  hypothesis,  is  p , and  let  it  be  re- 
quired to  determine  the  probability  that  if  the  phenomenon  y is 
observed,  the  hypothesis  x is  time.  The  very  data  of  this  pro- 
blem cannot  be  expressed  without  the  introduction  of  an  arbi- 
trary element.  W e can  only  write 

Prob.  x = a,  Prob.  xy  = ap;  (1) 

a being  perfectly  arbitrary,  except  that  it  must  fall  within  the 
limits  0 and  1 inclusive.  If  then  P represent  the  conditional  pro- 
bability sought,  we  have 

Prob.  xy  _ ap 

Prob.  y Prob.  y ^ 

It  remains  then  to  determine  Prob.  y. 


• Edinburgh  Review,  No.  185,  p.  32.  This  article,  though  not  entirely  free 
from  error,  is  well  worthy  of  attention. 


366 


PROBLEMS  ON  CAUSES. 


[chap.  XX. 


Let  xy  = t,  then 

y = t~x'=tX  + \t(l~  *)  + °0  + (!-«)•  (3) 


Hence  observing  that  Prob.  x = a,  Prob.  t = ap,  and  passing  from 
Logic  to  Algebra,  we  have 


-r,  , tx  + c(\-t)  x 

Prob.  y = — - — 

J tx  + 1 - t 


with  the  relations 


tx  + ( 1 - t)  x tx 
a ap 


= tx  + 1 - t. 


Hence  we  readily  find 

Prob.  y = ap  + c (1  - a).  (4) 

Now  recurring  to  (3),  we  find  that  c is  the  probability,  that  if 
the  event  (1  - t)  (1  - x)  occur,  the  event  y will  occur.  But 

(1  — #)  (1  — a?)  = (1  — xy)  (1  - x)  = 1 - x. 

Hence  c is  the  probability  that  if  the  event  x do  not  occur , 
the  event  y will  occur. 

Substituting  the  value  of  Prob.  y in  (2),  we  have  the  follow- 
ing theorem : 

The  calculated  probability  of  any  phcenomenon  y,  upon  an  as- 
sumed physical  hypothesis  x,  being  p,  the  a posteriori  probability  P 
of  the  physical  hypothesis,  when  the  phcenomenon  has  been  observed, 
is  expressed  by  the  equation 

P= ^ ,,  (5) 

ap  + c(l  - a)  v ' 


where  a and  c are  arbitrary  constants , the  former  representing  the 
d priori  probability  of  the  hypothesis , the  latter  the  probability  that 
if  the  hypothesis  were  false,  the  event  y would  present  itself. 

The  principal  conclusion  deducible  from  the  above  theorem 
is  that,  other  things  being  the  same,  the  value  of  P increases  and 
diminishes  simultaneously  with  that  of  p.  Hence  the  greater  or 
less  the  probability  of  the  phenomenon  when  the  hypothesis  is 
assumed,  the  greater  or  less  is  the  probability  of  the  hypothesis 
when  the  phcenomenon  has  been  observed.  When  p is  very  small, 
then  generally  P also  is  small,  unless  either  a is  large  or  c small. 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


367 


Hence,  secondly,  if  the  probability  of  tbe  phenomenon  is  very 
small  when  the  hypothesis  is  assumed,  the  probability  of  the  hy- 
pothesis is  very  small  when  the  phenomenon  is  observed,  unless 
either  the  a priori  probability  a of  the  hypothesis  is  large,  or  the 
probability  of  the  phenomenon  upon  any  other  hypothesis  small. 

The  formula  (5)  admits  of  exact  verification  in  various  cases, 
as  when  c = 0,  or  a = 1,  or  a = 0.  But  it  is  evident  that  it  does 
not,  unless  there  be  means  for  determining  the  values  of  a and  c, 
yield  a definite  value  of  P.  Any  solutions  which  profess  to  ac- 
complish this  object,  either  are  erroneous  in  principle,  or  involve 
a tacit  assumption  respecting  the  above  arbitrary  elements.  Mr. 
De  Morgan’s  solution  of  Laplace’s  problem  concerning  the  ex- 
istence of  a determining  cause  of  the  narrow  limits  within  which 
the  inclinations  of  the  planetary  orbits  to  the  plane  of  the  ecliptic 
are  confined,  appears  to  me  to  be  of  the  latter  description.  Having 
found  a probability  p = .00000012,  that  the  sum  of  the  incli- 
nations would  be  less  than  92°  were  all  degrees  of  inclination 
equally  probable  in  each  orbit,  this  able  writer  remarks : “ If 
there  be  a reason  for  the  inclinations  being  as  described,  the 
probability  of  the  event  is  1.  Consequently,  it  is  1 : .00000012 
(i.  e.  1 :p),  that  there  was  a necessary  cause  in  the  formation  of 
the  solar  system  for  the  inclinations  being  what  they  are.”  Now 
this  result  is  what  the  equation  (5)  would  really  give,  if,  assigning 


to  p the  above  value,  we  should  assume  c - 


should  thus  find, 


P 

1 + p 


For  we 


.-.  1 - P : P : : 1 : p.  (6) 

But  P representing  the  probability,  a posteriori,  that  all 
inclinations  are  equally  probable,  1 - P is  the  probability,  a pos- 
teriori, that  such  is  not  the  case,  or,  adopting  Mr.  De  Morgan’s 
alternative,  that  a determining  cause  exists.  The  equation  (6), 
therefore,  agrees  with  Mr.  De  Morgan’s  result. 

22.  Are  we,  however,  justified  in  assigning  to  a and  c parti- 
cular values?  I am  strongly  disposed  to  think  that  we  are  not. 


368 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


The  question  is  of  less  importance  in  the  special  instance  than 
in  its  ulterior  bearings.  In  the  received  applications  of  the  theory 
of  probabilities,  arbitrary  constants  do  not  explicitly  appear ; 
but  in  the  above,  and  in  many  other  instances  sanctioned  by  the 
highest  authorities,  some  virtual  determination  of  them  has  been 
attempted.  And  this  circumstance  has  given  to  the  results  of 
the  theory,  especially  in  reference  to  questions  of  causation,  a 
character  of  definite  precision,  which,  while  on  the  one  hand  it 
has  seemed  to  exalt  the  dominion  and  extend  the  province  of 
numbers,  even  beyond  the  measure  of  their  ancient  claim  to  rule 
the  world  ;*  on  the  other  hand  has  called  forth  vigorous  protests 
against  their  intrusion  into  realms  in  which  conjecture  is  the  only 
basis  of  inference.  The  very  fact  of  the  appearance  of  arbitrary 
constants  in  the  solutions  of  problems  like  the  above,  treated 
by  the  method  of  this  work,  seems  to  imply,  that  definite  solution 
is  impossible,  and  to  mark  the  point  where  inquiry  ought  to  stop. 
W e possess  indeed  the  means  of  interpreting  those  constants,  but 
the  experience  which  is  thus  indicated  is  as  much  beyond  our 
reach  as  the  experience  which  would  preclude  the  necessity  of 
any  attempt  at  solution  whatever. 

Another  difficulty  attendant  upon  these  questions,  and  inhe- 
rent, perhaps,  in  the  very  constitution  of  our  faculties,  is  that  of 
precisely  defining  what  is  meant  by  Order.  The  manifestations 
of  that  principle,  except  in  very  complex  instances,  we  have  no 
difficulty  in  detecting,  nor  do  we  hesitate  to  impute  to  it  an  al- 
most necessary  foundation  in  causes  operating  under  Law.  But 
to  assign  to  it  a standard  of  numerical  value  would  be  a vain, 
not  to  say  a presumptuous,  endeavour.  Yet  must  the  attempt  be 
made,  before  we  can  aspire  to  weigh  with  accuracy  the  probabi- 
bilities  of  different  constitutions  of  the  universe,  so  as  to  deter- 
mine the  elements  upon  which  alone  a definite  solution  of  the 
problems  in  question  can  be  established. 

23.  The  most  usual  mode  of  endeavouring  to  evade  the  ne- 
cessary arbitrariness  of  the  solution  of  problems  in  the  theory  of 


* Mundum  regunt  numcri. 

t See  an  interesting  paper  by  Prof.  Forbes  in  the  Philosophical  Magazine, 
Dec.  18.50;  also  Mill’s  Logic,  chap,  xviii. 


PROBLEMS  ON  CAUSES. 


369 


CHAP.  XX.] 

probabilities  which  rest  upon  insufficient  data,  is  to  assign  to  some 
element  whose  real  probability  is  unknown  all  possible  degrees 
of  probability ; to  suppose  that  these  degrees  of  probability  are 
themselves  equally  probable ; and,  regarding  them  as  so  many  dis- 
tinct causes  of  the  phenomenon  observed,  to  apply  the  theorems 
which  represent  the  case  of  an  effect  due  to  some  one  of  a number 
of  equally  probable  but  mutually  exclusive  causes  (Problem  9). 
For  instance,  the  rising  of  the  sun  after  a certain  interval  of 
darkness  having  been  observed  m times  in  succession,  the  proba- 
bility of  its  again  rising  under  the  same  circumstances  is  deter- 
mined, on  received  principles,  in  the  following  manner.  Let  p 
be  any  unknown  probability  between  0 and  1 , and  c (infinitesimal 
and  constant)  the  probability,  that  the  probability  of  the  sun’s 
rising  after  an  interval  of  darkness  lies  between  the  limits  p and 
p + dp.  Then  the  probability  that  the  sun  will  rise  m times  in 
succession  is 

e pmdp ; 

and  the  probability  that  he  will  do  this,  and  will  rise  again,  or, 
which  is  the  same  thing,  that  he  will  rise  m + 1 times  in  succes- 
sion, is 

c f pmtldp, 

Jo 

Hence  the  probability  that  if  he  rise  m times  in  succession,  he  will 
rise  the  m + 1 th  time,  is 

'If'Jp  m + l 

c (p-dp  m + 2' 

the  known  and  generally  received  solution. 

The  above  solution  is  usually  founded  upon  a supposed  analogy 
of  the  problem  with  that  of  the  drawing  of  balls  from  an  urn  con- 
taining a mixture  of  black  and  white  balls,  between  which  all 
possible  • numerical  ratios  are  assumed  to  be  equally  probable. 
And  it  is  remarkable,  that  there  are  two  or  three  distinct  hypo- 
theses which  lead  to  the  same  final  result.  For  instance,  if  the 
balls  are  finite  in  number,  and  those  which  are  drawn  arc  not 


370 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 

replaced,  or  if  they  are  infinite  in  number,  whether  those  drawn 
are  replaced  or  not,  then,  supposing  that  in  successive  drawings 
have  yielded  only  white  balls,  the  probability  of  the  issue  of  a 
white  ball  at  the  m + drawing  is 

m + 1 # 
m + 2 

It  has  been  said,  that  the  principle  involved  in  the  above 
and  in  similar  applications  is  that  of  the  equal  distribution  of 
our  knowledge,  or  rather  of  our  ignorance — the  assigning  to 
different  states  of  things  of  which  we  know  nothing,  and  upon 
the  very  ground  that  we  know  nothing,  equal  degrees  of  proba- 
bility. I apprehend,  however,  that  this  is  an  arbitrary  method  of 
procedure.  Instances  may  occur,  and  one  such  has  been  adduced, 
in  which  different  hypotheses  lead  to  the  same  final  conclusion. 
But  those  instances  are  exceptional.  With  reference  to  the  par- 
ticular problem  in  question,  it  is  shown  in  the  memoir  cited,  that 
there  is  one  hypothesis,  viz.,  when  the  balls  are  finite  in  number 
and  not  replaced,  which  leads  to  a different  conclusion,  and  it  is 
easy  to  see  that  there  are  other  hypotheses,  as  strictly  involving 
the  principle  of  the  “ equal  distribution  of  knowledge  or  igno- 
rance,” which  would  also  conduct  to  conflicting  results. 

24.  For  instance,  let  the  case  of  sunrise  be  represented  by 
the  drawing  of  a white  ball  from  a bag  containing  an  infinite 
number  of  balls,  which  are  all  either  black  or  white,  and  let  the 
assumed  principle  be,  that  all  possible  constitutions  of  the  system 
of  balls  are  equally  probable.  By  a constitution  of  the  system,  I 
mean  an  arrangement  which  assigns  to  every  ball  in  the  system 
a determinate  colour,  either  black  or  white.  Let  us  thence  seek 
the  probability,  that  if  m white  balls  are  drawn  in  m drawings, 
a white  ball  will  be  drawn  in  the  m + 1th  drawing. 

First,  suppose  the  number  of  the  balls  to  be  y,  and  let  the 
symbols  *l5  x2,  . . be  appropriated  to  them  in  the  following 
manner.  Let  denote  that  event  which  consists  in  the  i,h  ball 
of  the  system  being  white,  the  proposition  declaratory  of  such  a 
state  of  things  being  Xi  = 1.  In  like  manner  the  compound 


See  a memoir  by  Bishop  Terrot,  Edinburgh  Phil.  Trans,  vol.xx.  Part  iv. 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


371 


symbol  1 - xi  will  represent  the  circumstance  of  the  ith  ball  being 
black.  It  is  evident  that  the  several  constituents  formed  of  the 
entire  set  of  symbols  xly  x2,  . . will  represent  in  like  manner 
the  several  possible  constitutions  of  the  system  of  balls  with 
respect  to  blackness  and  whiteness,  and  the  number  of  such  con- 
stitutions being  2M,  the  probability  of  each  will,  in  accordance 

with  the  hypothesis,  be  ^ . This  is  the  value  which  we  should 

find  if  we  substituted  in  the  expression  of  any  constituent  for 

each  of  the  symbols  xu  x2,  . . x^,  the  value  Hence,  then,  the 

probability  of  any  event  which  can  be  expressed  as  a series  of 
constituents  of  the  above  description,  will  be  found  by  substi- 
tuting in  such  expression  the  value  ^ for  each  of  the  above 
symbols. 

Now  the  larger  ju  is,  the  less  probable  it  is  that  any  ball 
which  has  been  drawn  and  replaced  will  be  drawn  again.  As  fi 
approaches  to  infinity,  this  probability  approaches  to  0.  And 
this  being  the  case,  the  state  of  the  balls  actually  drawn  can  be 
expressed  as  a logical  function  of  m of  the  symbols  xlt . . x2  . . a^, 
and  therefore,  by  development,  as  a series  of  constituents  of  the 
said  m symbols.  Hence,  therefore,  its  probability  will  be  fonnd 
by  substituting  for  each  of  the  symbols,  whether  in  the  unde- 

1 

veloped  or  the  developed  form,  the  value  - . But  this  is  the  very 

£ 

substitution  which  it  would  be  necessary,  and  which  it  would 
suffice,  to  make,  if  the  probability  of  a white  ball  at  each  drawing 

were  known,  d priori,  to  be  ^ . 

JU 

It  follows,  therefore,  that  if  the  number  of  balls  be  infinite, 
and  all  constitutions  of  the  system  equally  probable,  the  proba- 
bility of  drawing  m white  balls  in  succession  will  be  and  the 


probability  of  drawing  m + 1 white  balls  in  succession 


2m+i  ’ 


whence  the  probability  that  after  m white  balls  have  been  drawn, 
the  next  drawing  will  furnish  a white  one,  will  be  In  other 


PROBLEMS  ON  CAUSES. 


372 


[chap.  XX. 


words,  past  experience  does  not  in  this  case  affect  future  ex- 
pectation. 

25.  It  may  be  satisfactory  to  verify  this  result  by  ordinary 
methods.  To  accomplish  this,  we  shall  seek — 

First : The  probability  of  drawing  r white  balls,  and  p - r 
black  balls,  in  p trials,  out  of  a bag  containing  p balls,  every  ball 
being  replaced  after  drawing,  and  all  constitutions  of  the  systems 
being  equally  probable,  d priori. 

Secondly : The  value  which  this  probability  assumes  when 
p becomes  infinite. 

Thirdly : The  probability  hence  derived,  that  if  m white 
balls  are  drawn  in  succession,  the  m + Ith  ball  drawn  will  be 
white  also. 

The  probability  that  r white  balls  and  p - r black  ones  will  be 
drawn  in  p trials  out  of  an  urn  containing  ju  balls,  each  ball 
being  replaced  after  trial,  and  all  constitutions  of  the  system  as 
above  defined  being  equally  probable,  is  equal  to  the  sum  of  the 
probabilities  of  the  same  result  upon  the  separate  hypotheses  of 
there  being  no  white  balls,  1 white  ball, — lastly  p white  balls  in 
the  urn.  Therefore,  it  is  the  sum  of  the  probabilities  of  this  re- 
sult on  the  hypothesis  of  there  being  n white  balls,  n varying 
from  0 to  p. 

Now  supposing  that  there  are  n white  balls,  the  probability 

of  drawing  a white  ball  in  a single  drawing  is  - , and  the  proba- 

bility  of'  drawing  r white  balls  and  p - r black  ones  in  a parti- 
cular order  in  p drawings,  is 

(;)'(' 

But  there  being  as  many  such  orders  as  there  are  combinations 
of  r things  in  p things,  the  total  probability  of  drawing  r white 
balls  in  p drawings  out  of  the  system  of  p balls  of  which  n are 
white,  is 

(l) 

Again,  the  number  of  constitutions  of  the  system  of  p balls,  which 
admit  of  exactly  n balls  being  white,  is 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


373 


n(n-l)  . . (ji-n+l) 

1.2  ..n 

and  the  number  of  possible  constitutions  of  the  system  is  2M. 
Hence  the  probability  that  exactly  n balls  are  white  is 

p(r*~  1)  • • (a*  - n+  1) 

1.2..  n 2" 

Multiplying  (1)  by  this  expression,  and  taking  the  sum  of  the 
products  from  n = 0 to  n = n,  we  have 


p(p-\)..p-r+\ 
1.2  ..r 


n = 0 


n (ju  - 1) . . ju  - « + i/»y/ 

1.2..W2'*  Vi“/V 


> (2) 


for  the  expression  of  the  total  probability,  that  out  of  a system 
of  fi  balls  of  which  all  constitutions  are  equally  probable,  r white 
balls  will  issue  in  p drawings.  Now 


..(//-»  + 1) 
n- o 1 . 2 . . n . 2* 


»=m  . . (p-n+i)  / »y, 

»--o  1 . 2 . . n 2M  U/' 


..(0  = 0) 


i z>v-^"=>(M-i)..(^-n  + i) 

s"*“  rraTT; 


£ 


(1+6^, 


(3) 


D standing  for  the  symbol  so  that  <p  (D)  en6=  <p  (w)  tnS . But 
by  a known  theorem, 

* A20m  A30m 

a!  Oire 

.-.  Z)m  (1  + £»),1  = {1  + A0mI>  +- — — D(D-  1)  + &c.)  (1  + t9y. 

1 . u 

In  the  second  member  let  ee  = x,  then 
* d A2  0m  d1 

Lr  (1  + tey  = (1  + A0m£—  + — + + X)M’ 

since 

* 

ERRATA.  — 3,  5,  and  6 from  bottom,  for  1 read  0m. 


374 


PROBLEMS  ON  CAUSES. 


[CHAP.  XX. 


In  the  second  member  of  the  above  equation,  performing  the  dif- 
ferentiations and  making  x = 1 (since  0 = 0),  we  get 

Dm  (1  +£ey  = n (A0m)  2m-1  + — (A20m)  2'1"2  + &c. 

I • A 

The  last  term  of  the  second  member  of  this  equation  will  be 
n(ii- 1)  • • l)Am0" 


1.2 ..m 


= ..(p-m+1)  2M_m; 


since  Am0wl  = 1 . 2 . . m.  When  p is  a large  quantity  this  term 
exceeds  all  the  others  in  value,  and  as  fi  approaches  to  infinity 
tends  to  become  infinitely  great  in  comparison  with  them.  And 
as  moreover  it  assumes  the  form  2U  ~ m,  we  have,  on  passing  to 

the  limit, 

Dm  (l  + t9y  = = f ^ ) 2m. 


Hence  if  $ (D)  represent  any  function  of  the  symbol  D,  which 
is  capable  of  being  expanded  in  a series  of  ascending  powers  of  D, 
we  have 

*(D)  (!  + .•>- *(!)»■,  (4) 


if  0 = 0 and  ju  = oo.  Strictly  speaking,  this  implies  that  the  ratio  of 
the  two  members  of  the  above  equation  approaches  a state  of 
equality,  as  n increases  towards  infinity,  0 being  equal  to  0. 

By  means  of  this  theorem,  the  last  member  of  (3)  reduces  to 
the  form 


1 

2m 


Hence  (2)  gives 

p (p  - 1)  . . (p  - r + 1 / 1 V 
1 .2..r  \2)  5 


as  the  expression  for  the  probability  that  from  an  urn  containing 
an  infinite  number  of  black  and  white  balls,  all  constitutions  of 
the  system  being  equally  probable,  r white  balls  will  issue  in  p 
drawings. 

Hence,  making  p = m,r  = m,  the  probability  that  in  m drawings 


all  the  balls  will  be  white 


is 


/l\m 

( - j , and  the  pi’obability 


that  this 


CHAP.  XX.] 


PROBLEMS  ON  CAUSES. 


375 


will  be  the  case,  and  that  moreover  the  m + l'A  drawing  will 

yield  a white  ball  is  , whence  the  probability,  that  if  the 

first  m drawings  yield  white  balls  only,  the  m + \th  drawing  will 
also  yield  a white  ball,  is 


and  generally,  any  proposed  result  will  have  the  same  probability 
as  if  it  were  an  even  chance  whether  each  particular  drawing 
yielded  a white  or  a black  ball.  This  agrees  with  the  conclusion 
before  obtained. 

26.  These  results  only  illustrate  the  fact,  that  when  the  defect 
of  data  is  supplied  by  hypothesis,  the  solutions  will,  in  general, 
vary  with  the  nature  of  the  hypotheses  assumed ; so  that  the 
question  still  remains,  only  more  definite  in  form,  whether  the 
principles  of  the  theory  of  probabilities  serve  to  guide  us  in  the 
election  of  such  hypotheses.  I have  already  expressed  my  convic- 
tion that  they  do  not — a conviction  strengthened  by  other  reasons 
than  those  above  stated.  Thus,  a definite  solution  of  a problem 
having  been  found  by  the  method  of  this  work,  an  equally  de- 
finite solution  is  sometimes  attainable  by  the  same  method  when 
one  of  the  data,  suppose  Prob.  x =px  is  omitted.  But  I have  not 
been  able  to  discover  any  mode  of  deducing  the  second  solution 
from  the  first  by  integration , with  respect  to  p supposed  variable 
within  limits  determined  by  Chap.  xix.  This  deduction  would, 
however,  I conceive,  be  possible,  were  the  principle  adverted  to 
in  Art.  23  valid.  Still  it  is  with  diffidence  that  I express  my 
dissent  on  these  points  from  mathematicians  generally,  and  more 
especially  from  one  who,  of  English  writers,  has  most  fully  en- 
tered into  the  spirit  and  the  methods  of  Laplace ; and  I venture 
to  hope,  that  a question,  second  to  none  other  in  the  Theory  of 
Probabilities  in  importance,  will  receive  the  careful  attention 
which  it  deserves. 


J \m  + i 
2 ) 


1Y*  _ 1 
2/  “ 2 


376 


PROBABILITY  OF  JUDGMENTS. 


[CHAP.  XXI. 


CHAPTER  XXI. 

PARTICULAR  APPLICATION  OF  THE  PREVIOUS  GENERAL  METHOD 

TO  THE  QUESTION  OF  THE  PROBABILITY  OF  JUDGMENTS. 

1 • the  presumption  that  the  general  method  of  this  treatise 

for  the  solution  of  questions  in  the  theory  of  probabilities, 
has  been  sufficiently  elucidated  in  the  previous  chapters,  it  is  pro- 
posed here  to  enter  upon  one  of  its  practical  applications  selected 
out  of  the  wide  field  of  social  statistics,  viz.,  the  estimation  of  the 
probability  of  judgments.  Perhaps  this  application,  if  weighed 
by  its  immediate  results,  is  not  the  best  that  could  have  been 
chosen.  One  of  the  first  conclusions  to  which  it  leads  is  that  of 
the  necessary  insufficiency  of  any  data  that  experience  alone  can 
furnish,  for  the  accomplishment  of  the  most  important  object  of 
the  inquiry.  But  in  setting  clearly  before  us  the  necessity  of 
hypotheses  as  supplementary  to  the  data  of  experience,  and  in 
enabling  us  to  deduce  with  rigour  the  consequences  of  any  hy- 
pothesis which  may  be  assumed,  the  method  accomplishes  all 
that  properly  lies  within  its  scope.  And  it  may  be  remarked, 
that  in  questions  which  relate  to  the  conduct  of  our  own  species, 
hypotheses  are  more  justifiable  than  in  questions  such  as  those  re- 
ferred to  in  the  concluding  sections  of  the  previous  chapter.  Our 
general  experience  of  human  nature  comes  in  aid  of  the  scantiness 
and  imperfection  of  statistical  records. 

2.  The  elements  involved  in  problems  relating  to  criminal 
assize  are  the  following : — 

1st.  The  probability  that  a particular  member  of  the  jury 
Avill  form  a correct  opinion  upon  the  case. 

2nd.  The  probability  that  the  accused  party  is  guilty. 

3rd.  The  probability  that  he  will  be  condemned,  or  that  he 
will  be  acquitted. 

4th.  The  probability  that  his  condemnation  or  acquittal  will 
be  just. 

5th.  The  constitution  of  the  jury. 


CHAP.  XXI.]  PROBABILITY  OF  JUDGMENTS.  377 

6 th.  The  data  furnished  by  experience,  such  as  the  relative 
numbers  of  cases  in  which  unanimous  decisions  have  been  arrived 
at,  or  particular  majorities  obtained ; the  number  of  cases  in 
which  decisions  have  been  reversed  by  superior  courts,  &c. 

Again,  the  class  of  questions  under  consideration  may  be 
regarded  as  either  direct  or  inverse.  The  direct  questions  of  pro- 
bability are  those  in  which  the  probability  of  correct  decision 
for  each  member  of  the  tribunal,  or  of  guilt  for  the  accused 
party,  are  supposed  to  be  known  a priori , and  in  which  the  proba- 
bility of  a decision  of  a particular  kind,  or  with  a definite  majority, 
is  sought.  Inverse  problems  are  those  in  which,  from  the  data  fur- 
nished by  experience,  it  is  required  to  determine  some  element 
which,  though  it  stand  to  those  data  in  the  relation  of  cause  to 
effect,  cannot  directly  be  made  the  subject  of  observation;  as 
when  from  the  records  of  the  decisions  of  courts  it  is  required  to 
determine  the  probability  that  a member  of  a court  will  judge 
correctly.  To  this  species  of  problems,  the  most  difficult  and 
the  most  important  of  the  whole  series,  attention  will  chiefly  be 
directed  here. 

3.  There  is  no  difficulty  in  solving  the  direct  problems  re- 
ferred to  in  the  above  enumeration.  Suppose  there  is  but  one 
juryman.  Let  k be  the  probability  that  the  accused  person  is 
guilty;  x the  probability  that  the  juryman  will  form  a correct 
opinion ; X the  probability  that  the  accused  person  will  be  con- 
demned : then — 

kx  = probability  that  the  accused  party  is  guilty,  and  that  the 
juryman  judges  him  to  be  guilty. 

(l-A)(l-x)  = probability  that  the  accused  person  is  inno- 
cent, and  that  the  juryman  pronounces  him  guilty. 

Now  these  being  the  only  cases  in  which  a verdict  of  con- 
demnation can  be  given,  and  being  moreover  mutually  exclusive, 
we  have 

X = kx  + (1  - k)  (1  - x).  (1) 

In  like  manner,  if  there  be  n jurymen  whose  separate  proba- 
bilities of  correct  judgment  are  x1}  x2  . . x„ , the  probability  of  an 
unanimous  verdict  of  condemnation  will  be 

X = kXi  x2  . . xn  + (.1  - k)  (1  - Xi)  (1  - x2)  . . (1  - X*). 


378 


PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI. 

Whence,  if  the  several  probabilities  Xi , x2  . . xn  are  equal,  and  are 
each  represented  by  x,  we  have 

X = kxn  + (1  - k)  (1  - x)n.  (2) 

The  probability  in  the  latter  case,  that  the  accused  person  is  guilty, 
will  be 

kxn 

kxn  + (1  - K)  (1  - x)n 

All  these  results  assume,  that  the  events  whose  probabilities 
are  denoted  by  k,  r15  x2,  &c.,  are  independent,  an  assumption 
which,  however,  so  far  as  we  are  concerned,  is  involved  in  the 
fact  that  those  events  are  the  only  ones  of  which  the  probabilities 
are  given. 

The  probability  of  condemnation  by  a given  number  of  voices 
may  be  found  on  the  same  principles.  If  a jury  is  composed  of 
three  persons,  whose  several  probabilities  of  correct  decision  are 
x,  x,  of',  the  probability  X2  that  the  accused  person  will  be  de- 
clared guilty  by  two  of  them  will  be 

X2  = k [xrf  (1  - x')  + xx"  (1  - x’)  + x o£'  (1  - a:)} 

+ (1  - k)  {(1  - x)  (l-af)x"  + (l-x)  (1  -a/')  X + (1  - af)  (1  -aT)  x), 

which  if  x = x = x"  reduces  to 

3kx 2 (1  - x)  + 3 (1  - k)  x (1  - x)2. 

And  by  the  same  mode  of  reasoning,  it  will  appear  that  if 
Xi  represent  the  probability  that  the  accused  person  will  be  de- 
clared guilty  by  i voices  out  of  a jury  consisting  of  n persons, 
whose  separate  probabilities  of  correct  judgment  are  equal,  and 
represented  by  x,  then 

Xi  = n(n "I) “ ~ 1 X)  (fee* ( 1 - x)n'i  + (1  - ^"^(l -x)*}.  (3) 

If  the  probability  of  condemnation  by  a determinate  majority  a 
is  required,  we  have  simply 

i - a = n - i, 

7i  + a 


whence 


PROBABILITY  OF  JUDGMENTS. 


379 


CHAP.  XXI.] 

which  must  be  substituted  in  the  above  formula.  Of  course  a 
admits  only  of  such  values  as  make  i an  integer.  If  n is  even, 
those  values  are  0,  2,  4,  &c. ; if  odd,  1,  3,  5,  &c.,  as  is  otherwise 
obvious. 

The  probability  of  a condemnation  by  a majority  of  at  least  a 
given  number  of  voices  m,  will  be  found  by  adding  together  the 
following  several  probabilities  determined  as  above,  viz. : 

1st.  The  probability  of  a condemnation  by  an  exact  ma- 
jority m ; 

2nd.  The  probability  of  condemnation  by  the  next  greater 
majority  m + 2 ; 

and  so  on ; the  last  element  of  the  series  being  the  probability  of 
unanimous  condemnation.  Thus  the  probability  of  condemnation 
by  a majority  of  4 at  least  out  of  12  jurors,  would  be 

-Xg  + -X9  . . + -X"l25 

the  values  of  the  above  terms  being  given  by  (3)  after  making 
therein  n = 12. 

4.  When,  instead  of  a jury,  we  are  considering  the  case  of  a 
simple  deliberative  assembly  consisting  of  n persons,  whose  sepa- 
rate probabilities  of  correct  judgment  are  denoted  by  x,  the  above 
formulae  are  replaced  by  others,  made  somewhat  more  simple  by 
the  omission  of  the  quantity  k. 

The  probability  of  unanimous  decision  is 


X = xn  + (1  - x)n. 

The  probability  of  an  agreement  of  i voices  out  of  the  whole 
number  is 

Xi=  n(n-1)  ' ^n~[  + j)  {^(l-^-i  + a*-^!-®)*}.  (4) 


Of  this  class  of  investigations  it  is  unnecessary  to  give  any 
further  account.  They  have  been  pursued  to  a considerable  ex- 
tent by  Condorcet,  Laplace,  Poisson,  and  other  writers,  who 
have  investigated  in  particular  the  modes  of  calculation  and  re- 
duction which  are  necessary  to  be  employed  when  n and  i are 
large  numbers.  It  is  apparent  that  the  whole  inquiry  is  of  a very 
speculative  character.  The  values  of  x and  k cannot  be  deter- 


380 


PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI, 

mined  by  direct  observation.  We  can  only  presume  that  they 
must  both  in  general  exceed  the  value  that  the  former,  x,  must 

increase  with  the  progress  of  public  intelligence ; while  the  latter, 
k,  must  depend  much  upon  those  preliminary  steps  in  the  ad- 
ministration of  the  law  by  which  persons  suspected  of  crime  are 
brought  before  the  tribunal  of  their  country.  It  has  been  re- 
marked by  Poisson,  that  in  periods  of  revolution,  as  during  the 
Reign  of  Terror  in  France,  the  value  of  k may  fall,  if  account  be 

taken  of  political  offences,  far  below  the  limit  ^ . The  history  of 

Europe  in  days  nearer  to  our  own  would  probably  confirm  this 
observation,  and  would  show  that  it  is  not  from  the  wild  license 
of  democracy  alone,  that  the  accusation  of  innocence  is  to  be 
apprehended. 

Laplace  makes  the  assumption,  that  all  values  of  x from 
x = ^,  to  x = 1, 

are  equally  probable.  He  thus  excludes  the  supposition  that  a 
juryman  is  more  likely  to  be  deceived  than  not,  but  assumes  that 
within  the  limits  to  which  the  probabilities  of  individual  cor- 
rectness of  judgment  are  confined,  we  have  no  reason  to  give 
preference  to  one  value  of  x over  another.  This  hypothesis  is 
entirely  arbitrary,  and  it  would  be  unavailing  here  to  examine 
into  its  consequences. 

Poisson  seems  first  to  have  endeavoured  to  deduce  the  values 
of  x and  k,  inferentially,  from  experience.  In  the  six  years  from 
1825  to  1830  inclusively,  the  number  of  individuals  accused  of 
crimes  against  the  person  before  the  tribunals  of  France  was 
11016,  and  the  number  of  persons  condemned  was  5286.  The 
juries  consisted  each  of  12  persons,  and  the  decision  was  pro- 
nounced by  a simple  majority.  Assuming  the  above  numbers 
to  be  sufficiently  large  for  the  estimation  of'  probabilities,  there 

5286 

would  therefore  be  a probability  measured  by  the  fraction  — — 

or  .4782  that  an  accused  person  would  be  condemned  by  a simple 
majority.  We  should  have  the  equation 

X-,  + A6  . . + -X"12  = .4782, 


(5) 


CHAP.  XXI.] 


PROBABILITY  OF  JUDGMENTS. 


381 


the  general  expression  for  Xt  being  given  by  (3)  after  making 
therein  n - 12.  In  the  year  1831  the  law,  having  received  alte- 
ration, required  a majority  of  at  least  four  persons  for  condemna- 
tion, and  the  number  of  persons  tried  for  crimes  against  the 
person  during  that  year  being  2046,  and  the  number  condemned 
7 43,  the  probability  of  the  condemnation  of  an  individual  by  the 
743 

above  majority  was  or  .3631.  Hence  we  should  have 

X8  + X9  . . . + X12  = .3631.  (6) 


Assuming  that  the  values  of  k and  x were  the  same  for  the 
year  1831  as  for  the  previous  six  years,  the  two  equations  (5)  and 
(6)  enable  us  to  determine  approximately  their  values.  Poisson 
thus  found, 

& = .5354,  x = .6786. 


For  crimes  against  property  during  the  same  periods,  he 
found  by  a similar  analysis, 


k = .6744,  x - .7771. 


The  solution  of  the  system  (5)  (6)  conducts  in  each  case  to 
two  values  of  k,  and  to  two  values  of  x,  the  one  value  in  each 


pair  being]  greater,  and  the  other  less,  than  - . It  was  assumed, 


that  in  each  case  the  larger  value  should  be  preferred,  it  being 
conceived  more  probable  that  a party  accused  , should  be  guilty 
than  innocent,  and  more  probable  that  a juryman  should  form 
a correct  than  an  erroneous  opinion  upon  the  evidence. 

5.  The  data  employed  by  Poisson,  especially  those  which  were 
furnished  by  the  year  1831,  are  evidently  too  imperfect  to  permit 
us  to  attach  much  confidence  to  the  above  determinations  of  x and 
k ; and  it  is  chiefly  for  the  sake  of  the  method  that  they  are  here 
introduced.  It  would  have  been  possible  to  record  during  the 
six  years,  1825-30,  or  during  any  similar  period,  the  number  of 
condemnations  pronounced  with  each  possible  majority  of  voices. 
The  values  of  the  several  elements  Xs,  X9,  . . X12,  were  there 
no  reasons  of  policy  to  forbid,  might  have  been  accurately  ascer- 
tained. Here  then  the  conception  of  the  general  problem,  of 
which  Poisson’s  is  a particular  case,  arises.  How  shall  we,  from 


382 


PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI. 

this  apparently  supernumerary  system  of  data,  determine  the 
values  of  x and  k ? If  the  hypothesis,  adopted  by  Poisson  and 
all  other  writers  on  the  subject,  of  the  absolute  independence  of 
the  events  whose  probabilities  are  denoted  by  x and  k be  retained, 
we  should  be  led  to  form  a system  of  five  equations  of  the  type  (3), 
and  either  select  from  these  that  particular  pair  of  equations  which 
might  appear  to  be  most  advantageous,  or  combine  together  the 
equations  of  the  system  by  the  method  of  least  squares.  There 
might  exist  a doubt  as  to  whether  the  latter  method  would  be 
strictly  applicable  in  such  cases,  especially  if  the  values  of  x and  k 
afforded  by  different  selected  pairs  of  the  given  equations  were  very 
different  from  each  other.  M.  Cournot  has  considered  a somewhat 
similar  problem,  in  which,  from  the  records  of  individual  votes  in 
a court  consisting  of  four  judges,  it  is  proposed  to  investigate  the 
separate  probabilities  of  a correct  verdict  from  each  judge.  For 
the  determination  of  the  elements  x,  x,  xf',  x",  he  obtains  eight 
equations,  which  he  divides  into  two  sets  of  four  equations,  and 
he  remarks,  that  should  any  considerable  discrepancy  exist  be- 
tween the  values  of  x , x',  x",  x",  determined  from  those  sets,  it 
might  be  regarded  as  an  indication  that  the  hypothesis  of  the  in- 
dependence of  the  opinions  of  the  judges  was,  in  the  particular 
case,  untenable.  The  principle  of  this  mode  of  investigation  has 
been  adverted  to  in  (XVIII.  4). 

6.  I proceed  to  apply  to  the  class  of  problems  above  indicated, 
the  method  of  this  treatise,  and  shall  inquire,  first,  whether  the 
records  of  courts  and  deliberative  assemblies,  alone,  can  furnish 
any  information  respecting  the  probabilities  of  correct  judgment 
for  their  individual  members,  and,  it  appearing  that  they  cannot, 
secondly,  what  kind  and  amount  of  necessary  hypothesis  will  best 
comport  with  the  actual  data. 

Proposition  I. 

From  the  mere  records  of  the  decisions  of  a court  or  deliberative 
assembly , it  is  not  possible  to  deduce  any  definite  conclusion  re- 
specting the  correctness  of  the  individual  judgments  of  its  members. 

Though  this  Proposition  may  appear  to  express  but  the  con- 
viction of  unassisted  good  sense,  it  will  not  be  without  interest  to 
show  that  it  admits  of  rigorous  demonstration. 


CHAP.  XXI.] 


PROBABILITY  OF  JUDGMENTS. 


383 


Let  us  suppose  the  case  of  a deliberative  assembly  consisting 
of  n members,  no  hypothesis  whatever  being  made  respecting 
the  dependence  or  independence  of  their  judgments.  Let  the 
logical  symbols  x1}  x2,  . , xn  be  employed  according  to  the  fol- 
lowing definition,  viz. : Let  the  generic  symbol  X;  denote  that 
event  which  consists  in  the  uttering  of  a correct  opinion  by  the 
ith  member,  Ai  of  the  court.  We  shall  consider  the  values  of 
Prob.  Xi,  Prob.  x2,  . . Prob.  xn,  as  the  qucesita  of  a problem,  the 
expression  of  whose  possible  data  we  must  in  the  next  place 
investigate. 

Now  those  data  are  the  probabilities  of  events  capable  of 
being  expressed  by  definite  logical  functions  of  the  symbols  x , , 
x2,  . .xn.  Let  Xi,  X2,  . . Xm  represent  the  functions  in  question, 
and  let  the  actual  system  of  data  be 

Prob.  Xx  = ax , Prob.  X2  = a2  Prob.  Xm  = am. 

Then  from  the  very  nature  of  the  case  it  may  be  shown  that 
.Xx,  X2,  . . Xm,  are  functions  which  remain  unchanged  if 
Xi,  x2,  . . xn  are  therein  changed  into  1 - xx , 1 — x2i  . . 1 - xn 
respectively.  Thus,  if  it  were  recorded  that  in  a certain  pro- 
portion of  instances  the  votes  given  were  unanimous,  the  event 
whose  probability,  supposing  the  instances  sufficiently  numerous, 
is  thence  determined,  is  expressed  by  the  logical  function 

X\  x2  . . xn  + (1  - (1  ~ x2 ) . . (1  — 

a function  which  satisfies  the  above  condition.  Again,  let  it  be 
recorded,  that  in  a certain  proportion  of  instances,  the  vote  of  an 
individual,  suppose  Ar,  differs  from  that  of  all  the  other  mem- 
bers of  the  court.  The  event,  whose  probability  is  thus  given, 
will  be  expressed  by  the  function 

x i (1  - x2)  . . (1  - xn)  + (1  - Xi)  x2  . . xn; 

also  satisfying  the  above  conditions.  Thus,  as  agreement  in 
opinion  may  be  an  agreement  in  either  truth  or  error ; and  as, 
when  opinions  are  divided,  either  party  may  be  right  or  wrong  ; 
it  is  manifest  that  the  expression  of  any  particular  state,  whether 
of  agreement  or  difference  of  sentiment  in  the  assembly,  will 
depend  upon  a logical  function  of  the  symbols  xlt  x2,  . . x„, 


384 


PROBABILITY  OF  JUDGMENTS. 


[CHAP.  XXI. 


which  similarly  involves  the  privative  symbols  1 - xXi  1 - x2, 
. . 1 - xn.  But  in  the  records  of  assemblies,  it  is  not  presumed 
to  declare  which  set  of  opinions  is  right  or  wrong.  Hence  the 
functions  Xx,  X2, . . Xm  must  be  solely  of  the  kind  above  de- 
scribed. 

7.  Now  in  proceeding,  according  to  the  general  method,  to 
determine  the  value  of  Prob.  we  should  first  equate  the  func- 

tions Xx,  . . Xm  to  a new  set  of  symbols  t . tm.  From  the 
equations 

X\  — t\)  X2  — t2,  . . Xm  — tm , 


thus  formed,  we  should  eliminate  the  symbols  x2 , x3,  . . xn,  and 
then  determine  xx  as  a developed  logical  function  of  the  symbols 
ti,  t.2,  . . tm,  expressive  of  events  whose  probabilities  are  given. 
Let  the  result  of  the  above  elimination  be 


Ex i + E'  (1  - Xi)  = 0 ; 

E and  E'  being  function  of  tlf  t2,  . . tm . Then 

E 

Xl~  E-E' 

Now  the  functions  Xx,  X2,  . . Xm  are  symmetrical  with  re- 
ference to  the  symbols  xx,  . . xn  and  1 - xx,  . . 1 -xn.  It  is  evi- 
dent, therefore,  that  in  the  equation  E must  be  identical  with  E. 
Hence  (2)  gives 

E 


0) 

(2) 


and  it  is  evident,  that  the  only  coefficients  which  can  appear  in.  the 
development  of  the  second  member  of  the  above  equation  are 

- and  -.  The  former  will  present  itself  whenever  the  values 
0 0 1 

assigned  to  tx , . . tm  in  determining  the  coefficient  of  a constituent, 
are  such  as  to  make  E = 0,  the  latter,  or  an  equivalent  result,  in 
every  other  case.  Hence  we  may  represent  the  development 
under  the  form 


*-Sc  + 5d' 


(3) 


C and  D being  constituents,  or  aggregates  of  constituents,  of  the 
symbols  tx,  t2,  . . tm. 


385 


CHAP.  XXI.]  PROBABILITY  OF  JUDGMENTS. 

Passing  then  from  Logic  to  Algebra,  we  have 

-p  i cC 
Prob.  ar,  = = c, 

the  function  V of  the  general  Rule  (XVII.  17)  reducing  in  the 
present  case  to  C.  The  value  of  Prob.  xx  is  therefore  wholly  ar- 
bitrary, if  we  except  the  condition  that  it  must  not  transcend 
the  limits  0 and  1.  The  individual  values  of  Prob.  x2, . . Prob.x„ , 
are  in  like  manner  arbitrary.  It  does  not  hence  follow,  that 
these  arbitrary  values  are  not  connected  with  each  other  by  ne- 
cessary conditions  dependent  upon  the  data.  The  investigation 
of  such  conditions  would,  however,  properly  fall  under  the  me- 
thods of  Chap.  xix. 

If,  reverting  to  the  final  logical  equation,  we  seek  the  inter- 
pretation of  c,  we  obtain  but  a restatement  of  the  original  pro- 
blem. For  since  C and  D together  include  all  possible  consti- 
tuents of  tx,  t2,  . . tm , we  have 

C + D = 1 ; 

and  since  D is  affected  by  the  coefficient  it  is  evident  that  on 

substituting  therein  for  tx , t2,  . . tm,  their  expressions  in  terms  of 
Xi,  x2,  . . x„ , we  should  have  D=  0.  Hence  the  same  substitution 
would  give  C = 1.  Now  by  the  rule,  c is  the  probability  that  if 
the  event  denoted  by  C take  place,  the  event  xx  will  take  place. 
Hence  C being  equal  to  1,  and,  therefore,  embracing  all  possible 
contingencies,  c must  be  interpreted  as  the  absolute  probability  of 
the  occurrence  of  the  event  Xi . 

It  may  be  interesting  to  determine  in  a particular  case  the 
actual  form  of  the  final  logical  equation.  Suppose,  then,  that  the 
elements  from  which  the  data  are  derived  are  the  records  of 
events  distinct  and  mutually  exclusive.  For  instance,  let  the 
numerical  data  a,,  a2,  . . am , be  the  respective  probabilities  of 
distinct  and  definite  majorities.  Then  the  logical  functions 
Xx,  X2,  . . Xm  being  mutually  exclusive,  must  satisfy  the  con- 
ditions 

Xx  X2  = 0,  . . Aj  Xm  = 0,  X2  Xm  — 0,  &c. 

Whence  we  have, 


A t-i  = 0,  tx  tm  = 0,  &c. 


386 


PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI. 


Under  these  circumstances  it  may  easily  be  shown,  that  the 
developed  logical  value  of  xx  will  be 

0, __  _ 

Xl  ~ 0 + ^ * * ^-0  j 


+ constitutents  whose  coefficients  are  - . 


In  the  above  equation  ~tx  stands  for  1 - tx , &c. 

These  investigations  are  equally  applicable  to  the  case  in 
which  the  probabilities  of  the  verdicts  of  a jury,  so  far  as  agree- 
ment and  disagreement  of  opinion  are  concerned,  form  the  data 
of  a problem.  Let  the  logical  symbol  w denote  that  event  or 
state  of  things  which  consists  in  the  guilt  of  the  accused  person. 
Then  the  functions  Xx,  X2  . . Xm  of  the  present  problem  are 
such,  that  no  change  would  therein  ensue  from  simultaneously 
converting  w,  xx,  x2  . . xn  into  w,  xx,  x2,  . . xn  respectively. 
Hence  the  final  logical  value  of  w,  as  well  as  those  o{xlt  x2,  . .xn 
will  be  exhibited  under  the  same  form  (3),  and  a like  general 
conclusion  thence  deduced. 

It  is  therefore  established,  that  from  mere  statistical  docu- 
ments nothing  can  be  inferred  respecting  either  the  individual 
correctness  of  opinion  of  a judge  or  counsellor,  the  guilt  of  an 
individual,  or  the  merits  of  a disputed  question.  If  the  deter- 
mination of  such  elements  as  the  above  can  be  reduced  within 
the  province  of  science  at  all,  it  must  be  by  virtue  either  of 
some  assumed  criterion  of  truth  furnishing  us  with  new  data,  or 
of  some  hypothesis  relative  to  the  connexion  or  the  independence 
of  individual  judgments,  which  may  warrant  a new  form  of  the 
investigation.  In  the  examination  of  the  results  of  different 
hypotheses,  the  following  general  Proposition  will  be  of  im- 
portance. 


Proposition  II. 

8.  Given  the  probabilities  of  the  n simple  events  xl}  x2,  . . xn, 
viz. : — 

Prob.  xx  = c,,  Prob.  x2  = c2,  . . Prob.  xn  - c„;  (1) 


also  the  probabilities  of  the  m - 1 compound  events  Xx , X2,  . . X,,^ , , 
viz. : — 

Prob.  - a ! , Prob.  X2  - a2,  . . Prob.  Xm . , = a„,.x ; (2) 


CHAP.  XXI.] 


PROBABILITY  OF  JUDGMENTS. 


387 


the  latter  events  Xx  . . Xm.x  being  distinct  and  mutually  exclusive ; 
required  the  'probability  of  any  other  compound  event  X. 

In  this  proposition  it  is  supposed,  that  Ax,  X2,  . . Xm_lt  as 
■well  as  X,  are  functions  of  the  symbols  xu  x2,  . . xn  alone. 
Moreover,  the  events  A1}  X2,  . . Xm_lf  being  mutually  exclusive, 
we  have 

Ai  X2  = 0,  . . Xi  Xm-i  = 0,  X2  A3  = 0,  &c. ; (3) 

the  product  of  any  two  members  of  the  system  vanishing.  Now 
assume 

Xi  = ti,  Am_i  = tm_  j,  X — t.  (4) 

Then  t must  be  determined  as  a logical  function  of  xx,  . . x„, 

• ■ t,ji  - 1 * 

Now  by  (3), 


^2  — o,  ti  tm_i  — 0,  t2 12  — 0,  Ac.  5 (5) 


all  binary  products  of  £x,  . . tm_  x,  vanishing.  The  developed  ex- 
pression for  t can,  therefore,  only  involve  in  the  list  of  constitu- 
ents which  have  1 , 0,  or  ^ for  their  coefficients,  such  as  contain 
some  one  of  the  following  factors,  viz. : — 


t\  t2 


t\  t2 


t\ 


tm-2  tm-l  5 


(6) 


Ti  standing  for  1 - tx , &c.  It  remains  to  assign  that  portion  of 
each  constituent  which  involves  the  symbols  xx . . xn  ; together 
with  the  corresponding  coefficients. 

Since  Xi  = tt  (i  being  any  integer  between  1 and  m - 1 inclu- 
sive), it  is  evident  that 

A{  tx  . . tm _ j = 0, 


from  the  very  constitution  of  the  functions.  Any  constituent 
included  in  the  first  member  of  the  above  equation  would,  there- 
fore, have  ^ for  its  coefficient. 

Now  let 

Xm  = 1 - Xx  . . - Xm.  i ; (7) 

and  it  is  evident  that  such  constituents  as  involve  7X  . . 7m.x , as 
a factor,  and  yet  have  coefficients  of  the  form  1,  0,  or  must  be 


388 


PROBABILITY  OF  JUDGMENTS. 


[CHAP.  XXI. 


included  in  the  expression 

Xm  ti  . tm.  i . 

Now  X„,  may  be  resolved  into  two  portions,  viz.,  XXm  and 
(1  - X)  Xm,  the  former  being  the  sum  of  those  constituents  of 
Xm  which  are  found  in  X,  the  latter  of  those  which  are  not  found 
in  X.  It  is  evident  that  in  the  developed  expression  of  t,  which 
is  equivalent  to  X , the  coefficients  of  the  constituents  in  the 
former  portion  XXm  will  be  l,  while  those  of  the  latter  portion 
(1  - X)  Xm  will  be  0.  Hence  the  elements  we  have  now  con- 
sidered will  contribute  to  the  development  of  t the  terms 

XXm  t\  . . Tm.i  + 0(1-  X)  Xm  ti  . . tm.  i . 

Again,  since  Xi  = ti , while  X2  tx  = t2  = 0,  &c.,  it  is  evident 
that  the  only  constituents  involving  txl2 . . lm.\  as  a factor  which 

0 

have  coefficients  of  the  form  1,  0,  or  - , will  be  included  in  the  ex- 
pression 

Xi  ti~f2  . . tm_  i ; 

and  reasoning  as  before,  we  see  that  this  will  contribute  to  the  de- 
velopment of  t the  terms 

XXi  tiT-2  . . fj/i-i  + 0(1  — Xs)  X\  tit-2  . . tm.\ 

Proceeding  thus  with  the  remaining  terms  of  (6),  we  deduce 
for  the  final  expression  of  t, 

t = XXm  ti  . . tm-\  + + XXm-Ji.. 

tm-2  tm- 1 

+ 0(1-X)X*71..7„-1  + 0(l-X)X1f1F8..7._1  + &e.  (8) 

+ terms  whose  coefficients  are  i. 

In  this  expression  it  is  to  be  noted  that  XXm  denotes  the  sum 
of  those  constituents  which  are  common  to  X and  XM,  that  sum 
being  actually  given  by  multiplying  X and  Xm together,  according 
to  the  rules  of  the  calculus  of  Logic. 

In  passing  from  Logic  to  Algebra,  we  shall  represent  by 
(XXm)  what  the  above  product  becomes,  when,  after  effecting 
the  multiplication,  or  selecting  the  common  constituents,  we 
give  to  the  symbols  a?M  . . xn,  a quantitative  meaning. 


CHAP.  XXI.]  PROBABILITY  OF  JUDGMENTS. 


389 


With  this  understanding  we  shall  have,  by  the  general  Rule 
(XVII.  17), 

Prob.  t 

(XXm)  tj . . tm.x  + (XXi)  t\  t%  • • tm-x*  • + (^XXm_  l)  t\ .. 

= _ 


'>  (») 


V Xm  t\  • . tm-  j -}-  -Xi  t'2  * • tm . i • • + Xm _ i t\  • • tm-2  ^m-1  ( 1 0) 

whence  the  relations  determining  xx,  . . xn,  tx,  . . tm.x  will  be  of 
the  following  type  (i  varying  from  1 to  n), 

(xj  XJ)  tx  . . tm_x  + ( Xj  X\)  tx  tz  . . tm_ i . . + ( Xj  Am‘_i)  tx  . . tm. q tm_i 


Ci 

X\  t[  t'2  . . tm _ i Xm - l 

CL\  CL  m -1 


= v. 


(11) 


From  the  above  system  we  shall  next  eliminate  the  symbols 

t\  5 • • tm- 1 • 

We  have 

. - - Qi  V ~1  ~I  x am-\ v /1£,N 

* 1 ?2  • • tm- 1 — „ , *1  • • - 2 6n  - 1 ~ -,r  • (I") 

■JL-l  JL.m-1 

Substituting  these  values  in  (10),  we  find 

V — Xm  7X  . . tm- 1 + o.x  V . . t Cl m _j  1 . 

Hence, 

__  _ (1  - ax  . . - am_i)  V 

t\  . . tm-\  — '• 

-A-m 

Now  let 

a.m  =1  — ax  . . — am-i , (13) 

then  we  have 

— — F /, 

^1  • • - 1 y • ( 1 4 ) 

-A- 771 

Now  reducing,  by  means  of  (12)  and  (14),  the  equation  (9), 
and  the  equation  formed  by  equating  the  first  line  of  ( 1 1 ) to  the 
symbol  V ; writing  also  Prob.  X for  Prob.  t , we  have 


Ai  A 2 AWi 

d\  (&iX i)  CLm(xiXni) 

xT~ + x2  -,+  xj 

wherein  A’m  and  nm  are  given  by  (7)  and  (13) 


(15) 


— ci  > 


(16) 


390  PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI. 

These  equations  involve  the  direct  solution  of  the  problem 
under  consideration.  In  (16)  we  have  the  type  of  n equations 
(formed  by  giving  to  i the  values  1,  2,  . . n successively),  from 
which  the  values  of  Xi,  x2, . . xn,  will  be  found,  and  those  values 
substituted  in  (15)  give  the  value  of  Prob.  X as  a function  of 
the  constants  aM  cl5  &c. 

One  conclusion  deserving  of  notice,  which  is  deducible  from 
the  above  solution,  is,  that  if  the  probabilities  of  the  compound 
events  Xls  . . Xm_i,  are  the  same  as  they  would  be  were  the 
events  xx,  . . xn  entirely  independent,  and  with  given  probabi- 
lities Ci,  . . cn,  then  the  probability  of  the  event  X will  be  the 
same  as  if  calculated  upon  the  same  hypothesis  of  the  absolute 
independence  of  the  events  xx,  . . xn.  For  upon  the  hypothesis 
supposed,  the  assumption  xx  = c1}  xn  = cn,  in  the  quantitative 
system  would  give  X1  = a,,  Xm  = am,  whence  (15)  and  (16) 
Would  give 

Prob.  X = (XX.)  + (XXS)  . . + (XX.),  (17) 

( xi  Xi)  + ( xi  X2)  . . + (xi  Xm)  = Ci . (18) 

But  since  X,  + Xr2  . . + Xm  = 1,  it  is  evident  that  the  second 
member  of  (17)  will  be  formed  by  taking  all  the  constituents  that 
are  contained  in  X,  and  giving  them  an  algebraic  significance. 
And  a similar  remark  applies  to  (18).  Whence  those  equations 
respectively  give 

Prob.  X (logical)  = X (algebraic), 

Xi  — Ci . 

Wherefore,  if  X = 0 (xx,  x2,  ■ ■ xn ),  we  have 
Prob.  X = 0 (ci,  ..  cn), 
which  is  the  result  in  question. 

Hence  too  it  would  follow,  that  if  the  quantities  c15 . . cn 
were  indeterminate,  and  no  hypothesis  were  made  as  to  the 
possession  of  a mean  common  value,  the  system  (15)  (16)  would 
be  satisfied  by  giving  to  those  quantities  any  such  values, 
Xi , x2 , . . xn , as  would  satisfy  the  equations 

Xi  = fli  • . A..,  = O-m-l,  X — a, 
supposing  the  value  of  the  element  a,  like  the  values  of  a,,. 
to  be  given  by  experience. 


PROBABILITY  OF  JUDGMENTS. 


391 


CHAP.  XXI.] 

9.  Before  applying  the  general  solution  (15)  (16),  to  the 
question  of  the  probability  of  judgments,  it  will  be  convenient  to 
make  the  following  transformation.  Let  the  data  be 

X\  = Ci  ....  Xn  = C713 


Prob.  Xi  = a!  ....  Prob.  Xm_2  = am-2 ; 

and  let  it  be  required  to  determine  Prob.  X,n_15  the  unknown 
value  of  which  we  will  represent  by  am_l.  Then  in  (15)  and  ( 1 6) 
we  must  change 

X into  Xm-\ , Prob.  X into  fl„H, 

XHI_i  into  Xm_2,  <zm-i  into  am-2. 

X,„  into  X.m_i  + X_ ;n , dm  into  , 

with  these  transformations,  and  observing  that  (!„.,  Xr)  = 0, 
except  when  r = m - 1,  and  that  it  is  then  equal  to  Xm_l5  the 
equations  (15)  (16)  give 

_ (#771-  1 -f  Clui)  X m- 1 
#77/ - l — v y 3 


C;  = 


Cl\(Xi  Ai)  ^ #/7i-2  (%i  Awl_2)  ^ (#77i-l  + #771)  Xm_ 1 + A"m)  ^20^ 


A 777  - 1 + A 77/ 


Now  from  (19)  we  find 

Xm_ 1 

#771-  1 


-A.  77 

am 


-AT  m _ i + Aw 

#7tt- 1 #7/i 


by  virtue  of  which  the  last  term  of  (20)  may  be  reduced  to  the 
form 

#771-1  (XiXm  _i)  ^ #777-  (X‘l  Atjj) 

Xm_i  A7/i 


With  these  reductions  the  system  (17)  and  (18)  may  be  replaced 
by  the  following  symmetrical  one,  viz. : 


Y 

^*-771-1 


xm 

9 

#771 


(21) 


«i  (*i  X.)  _ «2  (afj  X2)  aM(xiXm)  /ooN 

v-  + v '•4  v “ c,:‘  v^/ 

Ai  A2  -A  771 

These  equations,  in  connexion  with  (7)  and  (13),  enable  11s  to 


392  PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI 

determine  a,,,.,  as  a function  gf  ct . . cn,  a, . . am.2,  the  numerical 
data  supposed  to  be  furnished  by  experience.  We  now  proceed 
to  their  application. 

Proposition  III. 

10.  Given  any  system  of  probabilities  drawn  from  recorded 
instances  of  unanimity,  or  of  assigned  numerical  majority  in  the 
decisions  of  a deliberative  assembly  ; required,  upon  a certain  deter- 
minate hypothesis,  the  mean  probability  of  correct  judgment  for  a 
member  of  the  assembly. 

In  what  way  the  probabilities  of  unanimous  decision  and  of 
specific  numerical  majorities  may  be  determined  from  experience, 
has  been  intimated  in  a former  part  of  this  chapter.  Adopting 
the  notation  of  Prop.  i.  we  shall  represent  the  events  whose  pro- 
babilities are  given  by  the  functions  X,,  X2, . . Xm_1.  It  has 
appeared  from  the  very  nature  of  the  case  that  these  events  are 
mutually  exclusive,  and  that  the  functions  by  which  they  are  re- 
presented are  symmetrical  with  reference  to  the  symbols  ar15  x2, . .xn. 
Those  symbols  we  continue  to  use  in  the  same  sense  as  in  Prop.  I., 
viz.,  by  xi  we  understand  that  event  which  consists  in  the  for- 
mation of  a correct  opinion  by  the  ith  member  of  the  assembly. 

Now  the  immediate  data  of  experience  are — 

Prob.  X i = a,,  Prob.  X2  = a2,  . . Prob.  Xm.2  = am_2,  (1) 
Prob.  Xm.x  = am_,.  (2) 

X2 . . Xm.  i being  functions  of  the  logical  symbols  xx, ..  xn  to  the 
probabilities  of  the  events  denoted  by  which,  we  shall  assign  the 
indeterminate  value  c.  Thus  we  shall  have 

Prob.  X\  = Prob.  x2 . . = Prob.  xn  = c.  (3) 

Now  it  has  been  seen,  Prop,  i.,  that  the  immediate  data(l) 
(2),  unassisted  by  any  hypothesis,  merely  conduct  us  to  a re- 
statement of  the  problem.  On  the  other  hand,  it  is  manifest  that 
if,  adopting  the  methods  of  Laplace  and  Poisson,  we  employ  the 
system  (3)  alone  as  the  data  for  the  application  of  the  method  of 
this  work,  finally  comparing  the  results  obtained  with  the  expe- 
rimental system  (1)  (2),  we  are  relying  wholly  upon  a doubtful 
hypothesis, — the  independence  of  individual  judgments.  But 


CHAP.  XXI.]  PROBABILITY  OF  JUDGMENTS.  393 


though  we  ought  not  wholly  to  rely  upon  this  hypothesis,  we 
cannot  wholly  dispense  with  it,  or  with  some  equivalent  substi- 
tute. Let  us  then  examine  the  consequences  of  a limited  inde- 
pendence of  the  individual  judgments ; the  conditions  of  limitation 
being  furnished  by  the  apparently  superfluous  data.  From  the 
system  (1)  (3)  let  us,  by  the  method  of  this  work,  determine 
Prob.  Xm.x , and,  comparing  the  result  with  (2),  determine  c. 
Even  here  an  arbitrary  power  of  selection  is  claimed.  But  it  is 
manifest  from  Prop.  i.  that  something  of  this  kind  is  unavoidable, 
if  we  would  obtain  a definite  solution  at  all.  As  to  the  principle 
of  selection,  I apprehend  that  the  equation  (2)  reserved  for  final 
comparison  should  be  that  which,  from  the  magnitude  of  its  nu- 
merical element  am . x , is  esteemed  the  most  important  of  the  pri- 
mary series  furnished  by  experience. 

Now,  from  the  mutually  exclusive  character  of  the  events 
denoted  by  the  functions  Xx,  X2 , . . Xn^x , the  concluding  equa- 
tions of  the  previous  proposition  become  applicable.  On  account 
of  the  symmetry  of  the  same  functions,  and  the  reduction  of  the 
system  of  values  denoted  by  c,-  to  a single  value  c,  the  equations 
represented  by  (22)  become  identical,  the  values  of  xx,  xt, . . xn 
become  equal,  and  may  be  replaced  by  a single  value  x,  and  we 
have  simply, 


Xm-l  X„ 

&m- 1 


(4) 


cii  (xX i)  a^^xX 2)  cijji  (x Xm) 

~~Xi  + ~X7~  ‘ ' + Xn 


(5) 


The  following  is  the  nature  of  the  solution  thus  indicated  : 

The  functions  Xx, . . Xm.x,  and  the  values  ax.  . am_,,  being 
given  in  the  data,  we  have  first, 

Xm  = 1 — X, . . - Xm.x, 

am=  !-«!••-  am- i- 

From  each  of  the  functions  Xx,  X2,  . . Xm  thus  given  or  de- 
termined, we  must  select  those  constituents  which  contain  a par- 
ticular symbol,  as  xx,  for  a factor.  This  will  determine  the  func- 
tions (xXx),  (xX2),  &c.,  and  then  in  all  the  functions  we  must 
change  xx,  x2,  . . x„  individually  to  x.  Or  we  may  regard  any 


394 


PROBABILITY  OF  JUDGMENTS. 


[chap.  XXI. 


algebraic  function  Xi  in  the  system  (4)  (5)  as  expressing  the 
probability  of  the  event  denoted  by  the  logical  function  Xi,  on 
the  supposition  that  the  logical  symbols  xl,  x2,  , ,xn  denote  in- 
dependent events  whose  common  probability  is  x.  On  the  same 
supposition  ( xXt ) would  denote  the  probability  of  the  concur- 
rence of  any  particular  event  of  the  series  xx,  x2,  . . xn  with  Xi. 
The  forms  of  Xi,  (o;Ar;),  &c.  being  determined,  the  equation  (4) 
gives  the  value  of  x,  and  this,  substituted  in  (5),  determines  the 
value  of  the  element  c required.  Of  the  two  values  which  its  so- 
lution will  offer,  one  being  greater,  and  the  other  less,  than  i,  the 
greater  one  must  be  chosen,  whensoever,  upon  general  conside- 
rations, it  is  thought  more  probable  that  a member  of  the  assembly 
will  judge  correctly,  than  that  he  will  judge  incorrectly. 

Here  then,  upon  the  assumed  principle  that  the  largest  of 
the  values  am.x  shall  be  reserved  for  final  comparison  in  the 
equation  (2),  we  possess  a definite  solution  of  the  problem  pro- 
posed. And  the  same  form  of  solution  remains  applicable  should 
any  other  equation  of  the  system,  upon  any  other  ground,  as  that 
of  superior  accuracy,  be  similarly  reserved  in  the  place  of  (2). 

1 1 . Let  us  examine  to  what  extent  the  above  reservation  has 
influenced  the  final  solution.  It  is  evident  that  the  equation  (5) 
is  quite  independent  of  the  choice  in  question.  So  is  likewise 
the  second  member  of  (4).  Had  we  reserved  the  function  Xx, 
instead  of  Xm.x,  the  equation  for  the  determination  of#  would 
have  been 


stituted  in  the  same  final  equation  (5).  We  know  that  were 
the  events  xx,  x>,  . . xn  really  independent,  the  equations  (4), 
(6),  and  all  others  of  which  they  are  types,  would  prove  equi- 
valent, and  that  the  value  of  x furnished  by  any  one  of  them 
would  be  the  true  value  of  c.  This  affords  a means  of  verifying 
(5).  For  if  that  equation  be  correct,  it  ought,  under  the  above 
circumstances,  to  be  satisfied  by  the  assumption  c = x.  In  other 
words,  the  equation 


(6) 


but  the  value  of  x thence  determined  would  still  have  to  be  sub- 


(7) 


CHAP.  XXI.]  PROBABILITY  OF  JUDGMENTS. 


395 


ought,  on  solution,  to  give  the  same  value  of  x as  the  equation 
(4)  or  (6).  Now  this  will  be  the  case.  For  since,  by  hypothesis, 

X1=X2  xm 

«1  a2  ” am  ’ 
we  have,  by  a known  theorem, 

X*  = X,  m m = = x1  + x2..+  xm  = 1 

«2  ’ * a-m  Oi  + a2  . . + am 

Hence  (7)  becomes  on  substituting  al  lor  A,,  &c. 

(xX,)  + (xX2) . . + (xXm)  = x 

a mere  identity. 

Whenever,  therefore,  the  events  xl}  x2,  . . xn  are  really  inde- 
pendent, the  system  (4)  (5)  is  a correct  one,  and  is  independent 
of  the  arbitrariness  of  the  first  step  of  the  process  by  which  it 
was  obtained.  When  the  said  events  are  not  independent,  the 
final  system  of  equations  will  possess,  leaving  in  abeyance  the 
principle  of  selection  above  stated,  an  arbitrary  element.  But 
from  the  persistent  form  of  the  equation  (5)  it  may  be  inferred 
that  the  solution  is  arbitrary  in  a less  degree  than  the  solutions 
to  which  the  hypothesis  of  the  absolute  independence  of  the  in- 
dividual judgments  would  conduct  us.  The  discussion  of  the 
limits  of  the  value  of  c,  as  dependent  upon  the  limits  of  the  value 
of  x,  would  determine  such  points. 

These  considerations  suggest  to  us  the  question  whether  the 
equation  (7),  which  is  symmetrical  with  reference  to  the  func- 
tions X1}  X2, . . Xm , free  from  any  arbitrary  elements,  and  rigo- 
rously exact  when  the  events  xx,  x2, . . xn  are  really  independent, 
might  not  be  accepted  as  a mean  general  solution  of  the  problem. 
The  proper  mode  of  determining  this  point  would,  I conceive,  be 
to  ascertain  whether  the  value  of  x which  it  would  afford  would, 
in  general,  fall  within  the  limits  of  the  value  of  c,  as  determined 
by  the  systems  of  equations  of  which  the  system  (4),  (5),  presents 
the  type.  It  seems  probable  that  under  ordinary  circumstances 
this  would  be  the  case.  Independently  of  such  considerations, 
however,  we  may  regard  (7)  as  itself  the  expression  of  a certain 
principle  of  solution,  viz.,  that  regarding  A',,  A\, . . Am  as  ex- 
clusive causes  of  the  event  whose  probability  is  x,  we  accept  the 


39G 


PROBABILITY  OF  JUDGMENTS.  [CHAP.  XXI. 


probabilities  of  those  causes  a„  . . am  from  experience,  but  form 
the  conditional  probabilities  of  the  event  as  dependent  upon  such 
causes, 


&c.  (XVII.  Prop,  i.) 

Ao 


on  the  hypothesis  of  the  independence  of  individual  judgments, 
and  so  deduce  the  equation  (7).  I conceive  this,  however,  to  be 
a less  rigorous,  though  possibly,  in  practice  a more  convenient 
mode  of  procedure  than  that  adopted  in  the  general  solution. 

12.  It  now  only  remains  to  assign  the  particular  forms  which 
the  algebraic  functions  Xi,  (xXi),  &c.  in  the  above  equations  as- 
sume when  the  logical  function  Xi  represents  that  event  which 
consists  in  r members  of  the  assembly  voting  one  way,  and  n-r 
members  the  other  way.  It  is  evident  that  in  this  case  the  alge- 
braic function  Xi  expresses  what  the  probability  of  the  supposed 
event  would  be  were  the  events  x„  x2,  . . xn  independent,  and 
their  common  probability  measured  by  x.  Hence  we  should 
have,  by  Art.  3, 


n (n  - 1) ..  (n  - r + 1) 

1.2  ..r 


+ (1  -x)n'rj. 


Under  the  same  circumstances  ( xXi ) would  represent  the  pro- 
bability of  the  compound  event,  which  consists  in  a particular 
member  of  the  assembly  forming  a correct  judgment,  conjointly 
with  the  general  state  of  voting  recorded  above.  It  would, 
therefore,  be  the  probability  that  a particular  member  votes  cor- 
rectly, while  of  the  remaining  n - 1 members,  r - 1 vote  cor- 
rectly ; or  that  the  same  member  votes  correctly,  while  of  the 
remaining  n - 1 members  r vote  incorrectly.  Hence 


(xXi)  = 


(n  -1)  (n  - 2) . . [n  - r + 1) 

1 . 2 . . r - 1 


• + fo-!)  (w-2)..(»-r)  ^ 
1 .2  . . r 


Proposition  IV. 


13.  Given  any  system  of  probabilities  drawn  from  recorded  in- 
stances of  unanimity , or  of  assigned  numerical  majority  in  the  de- 
cisions of  a criminal  court  of  justice , required  upon  hypotheses 
similar  to  those  of  the  last  proposition , the  mean  probability  c of 


PROBABILITY  OF  JUDGMENTS. 


397 


CHAP.  XXI.] 


correct  judgment  for  a member  of  the  court,  and  the  general  pro- 
bability k of  guilt  in  an  accused  person. 


The  solution  of  this  problem  differs  in  but  a slight  degree 
from  that  of  the  last,  and  may  be  referred  to  the  same  general 
formulae,  (4)  and  (5),  or  (7).  It  is  to  be  observed,  that  as  there 
are  two  elements,  c and  k,  to  be  determined,  it  is  necessary  to 
reserve  two  of  the  functions  Xu  X2, . . Xm.x,  let  us  suppose  A", 
and  Xm.i,  for  final  comparison,  employing  either  the  remaining 
m - 3 functions  in  the  expression  of  the  data,  or  the  two  respec- 
tive sets  X2,  X3,  . . Xm.u  and  Xx,  X2,  . . ■ Xm.2.  In  either  case 
it  is  supposed  that  there  must  be  at  least  two  original  indepen- 
dent data.  If  the  equation  (7)  be  alone  employed,  it  would  in 
the  present  instance  furnish  two  equations,  which  may  thus  be 
written : 


afxXj)  a2(xX2)  am(x  Xm) 

X , + x2  • • Xm 


0) 


afkXi)  a2(kX2) , , am{kXm)  7 

v + + v k 

-Ai  A2 


These  equations  are  to  be  employed  in  the  following  manner : — 
Let  xx,  x2,  . . x„  represent  those  events  which  consist  in  the  for- 
mation of  a correct  opinion  by  the  members  of  the  court  respec- 
tively. Let  also  w represent  that  event  which  consists  in  the 
guilt  of  the  accused  member.  By  the  aid  of  these  symbols  we 
can  logically  express  the  functions  Xx,  X2, . . Xm_1}  whose  proba- 
bilities are  given,  as  also  the  function  Xm.  Then  from  the  func- 
tion Xx  select  those  constituents  which  contain,  as  a factor,  any  . 
particular  symbol  of  the  set  xlf  x2, . .xn,  and  also  those  consti- 
tuents which  contain  as  a factor  w.  In  both  results  change 
xlf  x2,  . . xn  severally  into  x,  and  w into  k.  The  above  results 
will  give  (xXj)  and  (AAj).  Effecting  the  same  transformations 
throughout,  the  system  (1),  (2)  will,  upon  the  particular  hypo- 
thesis involved,  determine  x and  h. 

14.  We  may  collect  from  the  above  investigations  the  fol- 
lowing facts  and  conclusions : 

1st.  That  from  the  mere  records  of  agreement  and  disagree- 
ment in  the  opinions  of  any  body  of  men,  no  definite  numerical 
conclusions  can  be  drawn  respecting  either  the  probability  of  cor- 


PROBABILITY  OF  JUDGMENTS. 


398 


[chap.  XXI. 


rect  judgment  in  an  individual  member  of  the  body,  or  the  merit 
of  the  questions  submitted  to  its  consideration. 

2nd.  That  such  conclusions  may  be  drawn  upon  various  dis- 
tinct hypotheses,  as — 1st,  Upon  the  usual  hypothesis  of  the  abso- 
lute independence  of  individual  judgments ; 2ndly,  upon  certain 
definite  modifications  of  that  hypothesis  warranted  by  the  actual 
data ; 3rdly,  upon  a distinct  principle  of  solution  suggested  by 
the  appearance  of  a common  form  in  the  solutions  obtained  by 
the  modifications  above  adverted  to. 

Lastly.  That  whatever  of  doubt  may  attach  to  the  final  re- 
sults, rests  not  upon  the  imperfection  of  the  method,  which 
adapts  itself  equally  to  all  hypotheses,  but  upon  the  uncertainty 
of  the  hypotheses  themselves. 

It  seems,  however,  probable  that  with  even  the  widest  limits 
of  hypothesis,  consistent  with  the  taking  into  account  of  all  the 
data  of  experience,  the  deviation  of  the  results  obtained  would  be 
but  slight,  and  that  their  mean  values  might  be  determined  with 
great  confidence  by  the  methods  of  Prop.  hi.  Of  those  methods 
I should  be  disposed  to  give  the  preference  to  the  first.  Such  a 
principle  of  mean  solution  having  been  agreed  upon,  other  consi- 
derations seem  to  indicate  that  the  values  of  c and  k for  tribunals 
and  assemblies  possessing  a definite  constitution,  and  governed 
in  their  deliberations  by  fixed  rules,  would  remain  nearly  con- 
stant, subject,  however,  to  a small  secular  variation,  dependent 
upon  the  progress  of  knowledge  and  of  justice  among  mankind. 
There  exist  at  present  few,  if  any,  data  proper  for  their  determi- 
nation. 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT. 


399 


CHAPTER  XXII. 

ON  THE  NATURE  OF  SCIENCE,  AND  THE  CONSTITUTION  OF  THE 
INTELLECT. 

1*  TTTHAT  I mean  by  the  constitution  of  a system  is  the 
“ ' aggregate  of  those  causes  and  tendencies  which  pro- 
duce its  observed  character,  when  operating,  without  interference, 
under  those  conditions  to  which  the  system  is  conceived  to  be 
adapted.  Our  judgment  of  such  adaptation  must  be  founded 
upon  a study  of  the  circumstances  in  which  the  system  attains  its 
freest  action,  produces  its  most  harmonious  results,  or  fulfils  in 
some  other  way  the  apparent  design  of  its  construction.  There 
are  cases  in  which  we  know  distinctly  the  causes  upon  which  the 
operation  of  a system  depends,  as  well  as  its  conditions  and  its 
end.  This  is  the  most  perfect  kind  of  knowledge  relatively  to 
the  subject  under  consideration.  There  are  also  cases  in  which 
we  know  only  imperfectly  or  partially  the  causes  which  are  at 
work,  but  are  able,  nevertheless,  to  determine  to  some  extent 
the  laws  of  their  action,  and,  beyond  this,  to  discover  general 
tendencies,  and  to  infer  ulterior  purpose.  It  has  thus,  I think 
rightly,  been  concluded  that  there  is  a moral  faculty  in  our  na- 
ture, not  because  we  can  understand  the  special  instruments  by 
which  it  works,  as  we  connect  the  organ  with  the  faculty  of  sight, 
nor  upon  the  ground  that  men  agree  in  the  adoption  of  universal 
rules  of  conduct ; but  because  while,  in  some  form  or  other,  the 
sentiment  of  moral  approbation  or  disapprobation  manifests  itself 
in  all,  it  tends,  wherever  human  progress  is  observable,  wherever 
society  is  not  either  stationary  or  hastening  to  decay,  to  attach 
itself  to  certain  classes  of  actions,  consentaneously,  and  after  a 
manner  indicative  both  of  permanency  and  of  law.  Always  and 
everywhere  the  manifestation  of  Order  affords  a presumption,  not 
measurable  indeed,  but  real  (XX.  22),  of  the  fulfilment  of  an  end 
or  purpose,  and  the  existence  of  a ground  of  orderly  causation. 


400  CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

2.  The  particular  question  of  the  constitution  of  the  intellect 
has,  it  is  almost  needless  to  say,  attracted  the  efforts  of  speculative 
ingenuity  in  every  age.  For  it  not  only  addresses  itself  to  that 
desire  of  knowledge  which  the  greatest  masters  of  ancient  thought 
believed  to  be  innate  in  our  species,  but  it  adds  to  the  ordinary 
strength  of  this  motive  the  inducement  of  a human  and  personal 
interest.  A genuine  devotion  to  truth  is,  indeed,  seldom  partial 
in  its  aims,  but  while  it  prompts  to  expatiate  over  the  fair  fields  of 
outward  observation,  forbids  to  neglect  the  study  of  our  own  fa- 
culties. Even  in  ages  the  most  devoted  to  material  interests, 
some  portion  of  the  current  of  thought  has  been  reflected  in- 
wards, and  the  desire  to  comprehend  that  by  which  all  else  is 
comprehended  has  only  been  baffled  in  order  to  be  renewed. 

It  is  probable  that  this  pertinacity  of  effort  would  not  have 
been  maintained  among  sincere  inquirers  after  truth,  had  the 
conviction  been  general  that  such  speculations  are  hopelessly 
barren.  W e may  conceive  that  it  has  been  felt  that  if  something 
of  error  and  uncertainty,  always  incidental  to  a state  of  partial 
information,  must  ever  be  attached  to  the  results  of  such  in- 
quiries, a residue  of  positive  knowledge  may  yet  remain ; that 
the  contradictions  which  are  met  with  are  more  often  verbal  than 
real ; above  all,  that  even  probable  conclusions  derive  here  an  in- 
terest and  a value  from  their  subject,  which  render  them  not 
unworthy  to  claim  regard  beside  the  more  definite  and  more 
splendid  results  of  physical  science.  Such  considerations  seem 
to  be  perfectly  legitimate.  Insoluble  as  many  of  the  problems 
connected  with  the  inquiry  into  the  nature  and  constitution  of 
the  mind  must  be  presumed  to  be,  there  are  not  wanting  others 
upon  which  a limited  but  not  doubtful  knowledge,  others  upon 
which  the  conclusions  of  a highly  probable  analogy,  are  attain- 
able. As  the  realms  of  day  and  night  are  not  strictly  contermi- 
nous, but  are  separated  by  a crepuscular  zone,  through  which  the 
light  of  the  one  fades  gradually  off  into  the  darkness  of  the  other, 
so  it  may  be  said  that  every  region  of  positive  knowledge  lies  sur- 
rounded by  a debateable  and  speculative  territory,  over  which  it 
in  some  degree  extends  its  influence  and  its  light.  Thus  there 
may  be  questions  relating  to  the  constitution  of  the  intellect 
which,  though  they  do  not  admit,  in  the  present  state  of  know- 


CHAP.  XXXI.] 


CONSTITUTION  OF  THE  INTELLECT. 


401 


ledge,  of  an  absolute  decision,  may  receive  so  much  of  reflected 
information  as  to  render  their  probable  solution  not  difficult ; and 
there  may  also  be  questions  relating  to  the  nature  of  science,  and 
even  to  particular  truths  and  doctrines  of  science,  upon  which 
they  who  accept  the  general  principles  of  this  work  cannot  but  be 
led  to  entertain  positive  opinions,  differing,  it  may  be,  from  those 
which  are  usually  received  in  the  present  day.*  In  what  fol- 
lows I shall  recapitulate  some  of  the  more  definite  conclusions 
established  in  the  former  parts  of  this  treatise,  and  shall  then 
indicate  one  or  two  trains  of  thought,  connected  with  the  gene- 
ral objects  above  adverted  to,  which  they  seem  to  me  calculated 
to  suggest. 

3.  Among  those  conclusions,  relating  to  the  intellectual  con- 
stitution, which  may  be  considered  as  belonging  to  the  realm  of 
positive  knowledge,  we  may  reckon  the  scientific  laws  of  thought 
and  reasoning,  which  have  formed  the  basis  of  the  general  me- 
thods of  this  treatise,  together  with  the  principles,  Chap,  v.,  by 
which  their  application  has  been  determined.  The  resolution  of 
the  domain  of  thought  into  two  spheres,  distinct  but  coexistent 
(IV.  XI.) ; the  subjection  of  the  intellectual  operations  within 
those  spheres  to  a common  system  of  laws  (XI.);  the  general 
mathematical  character  of  those  laws,  and  their  actual  expression 
(II.  III.)  ; the  extent  of  their  affinity  with  the  laws  of  thought  in 
the  domain  of  number,  and  the  point  of  their  divergence  there- 
from ; the  dominant  character  of  the  two  limiting  conceptions  of 
universe  and  eternity  among  all  the  subjects  of  thought  with 
which  Logic  is  concerned ; the  relation  of  those  conceptions  to 
the  fundamental  conception  of  unity  in  the  science  of  number, — 
these,  with  many  similar  results,  are  not  to  be  ranked  as  merely 


* The  following  illustration  may  suffice  : — 

It  is  maintained  by  some  of  the  highest  modern  authorities  in  grammar  that 
conjunctions  connect  propositions  only.  Now,  without  inquiring  directly  whe- 
ther this  opinion  is  sound  or  not,  it  is  obvious  that  it  cannot  consistently  beheld 
by  any  who  admit  the  scientific  principles  of  this  treatise  ; for  to  such  it  would 
seem  to  involve  a denial,  either,  1st,  of  the  possibility  of  performing , or  2ndly,  of 
the  possibility  of  expressing , a mental  operation,  the  laws  of  which,  viewed  in 
both  these  relations,  have  been  investigated  and  applied  in  the  present  work — 
(Latham  on  the  English  Language;  Sir  John  Stoddart’s  Universal  Gram- 
mar, &c.) 


402 


CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

probable  or  analogical  conclusions,  but  are  entitled  to  be  re- 
garded as  truths  of  science.  Whether  they  be  termed  meta- 
physical or  not,  is  a matter  of  indifference.  The  nature  of  the 
evidence  upon  which  they  rest,  though  in  kind  distinct,  is  not 
inferior  in  value  to  any  which  can  be  adduced  in  support  of  the 
general  truths  of  physical  science. 

Again,  it  is  agreed  that  there  is  a certain  order  observ- 
able in  the  progress  of  all  the  exacter  forms  of  knowledge. 
The  study  of  every  department  of  physical  science  begins  with 
observation,  it  advances  by  the  collation  of  facts  to  a presump- 
tive acquaintance  with  their  connecting  law,  the  validity  of 
such  presumption  it  tests  by  new  experiments  so  devised  as  to 
augment,  if  the  presumption  be  well  founded,  its  probability  in- 
definitely ; and  finally,  the  law  of  the  phenomenon  having  been 
with  sufficient  confidence  determined,  the  investigation  of  causes, 
conducted  by  the  due  mixture  of  hypothesis  and  deduction, 
crowns  the  inquiry.  In  this  advancing  order  of  knowledge,  the 
particular  faculties  and  laws  whose  nature  has  been  considered 
in  this  work  bear  their  part.  It  is  evident,  therefore,  that  if  we 
would  impartially  investigate  either  the  nature  of  science,  or 
the  intellectual  constitution  in  its  relation  to  science,  no  part  of 
the  two  series  above  presented  ought  to  be  regarded  as  isolated. 
More  especially  ought  those  truths  which  stand  in  any  kind  of 
supplemental  relation  to  each  other  to  be  considered  in  their  mu- 
tual bearing  and  connexion. 

4.  Thus  the  necessity  of  an  experimental  basis  for  all  positive 
knowledge,  viewed  in  connexion  with  the  existence  and  the 
peculiar  character  of  that  system  of  mental  laws,  and  principles, 
and  operations,  to  which  attention  has  been  directed,  tends  to 
throw  light  upon  some  important  questions  by  which  the  world 
of  speculative  thought  is  still  in  a great  measure  divided.  How, 
from  the  particular  facts  which  experience  presents,  do  we  arrive 
at  the  general  propositions  of  science  ? What  is  the  nature  of 
these  propositions  ? Are  they  solely  the  collections  of  experi- 

ence, or  does  the  mind  supply  some  connecting  principle  of  its 
own?  In  a word,  what  is  the  nature  of  scientific  truth,  and 
what  are  the  grounds  of  that  confidence  with  which  it  claims  to 
be  received? 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT. 


403 


That  to  such  questions  as  the  above,  no  single  and  general 
answer  can  be  given,  must  be  evident.  There  are  cases  in  which 
they  do  not  even  need  discussion.  Instances  are  familiar,  in 
which  general  propositions  merely  express  per  enumerationem 
simplicem,  a fact  established  by  actual  observation  in  all  the 
cases  to  which  the  proposition  applies.  The  astronomer  as- 
serts upon  this  ground,  that  all  the  known  planets  move  from 
west  to  east  round  the  sun.  But  there  are  also  cases  in  which 
general  propositions  are  assumed  from  observation  of  their  truth 
in  particular  instances,  and  extension  of  that  truth  to  instances 
unobserved.  No  principle  of  merely  deductive  reasoning  can 
warrant  such  a procedure.  When  from  a large  number  of  ob- 
servations on  the  planet  Mars,  Kepler  inferred  that  it  revolved 
in  an  ellipse,  the  conclusion  was  larger  than  his  premises,  or  in- 
deed than  any  premises  which  mere  observation  could  give. 
What  other  element,  then,  is  necessary  to  give  even  a prospective 
validity  to  such  generalizations  as  this  ? It  is  the  ability  in- 
herent in  our  nature  to  appreciate  Order,  and  the  concurrent  pre- 
sumption, however  founded,  that  the  phenomena  of  Nature  are 
connected  by  a principle  of  Order.  Without  these,  the  general 
truths  of  physical  science  could  never  have  been  ascertained. 
Grant  that  the  procedure  thus  established  can  only  conduct  us 
to  probable  or  to  approximate  results  ; it  only  follows,  that  the 
larger  number  of  the  generalizations  of  physical  science  possess 
but  a probable  or  approximate  truth.  The  security  of  the  tenure 
of  knowledge  consists  in  this,  that  wheresoever  such  conclusions 
do  truly  represent  the  constitution  of  Nature,  our  confidence  in 
their  truth  receives  indefinite  confirmation,  and  soon  becomes 
undistinguishable  from  certainty.  The  existence  of  that  prin- 
ciple above  represented  as  the  basis  of  inductive  reasoning 
enables  us  to  solve  the  much  disputed  question  as  to  the  neces- 
sity of  general  propositions  in  reasoning.  The  logician  affirms, 
that  it  is  impossible  to  deduce  any  conclusion  from  particular 
premises.  Modern  writers  of  high  repute  have  contended,  that 
all  reasoning  is  from  particular  to  particular  truths.  They  in- 
stance, that  in  concluding  from  the  possession  of  a property  by 
certain  members  of  a class,  its  possession  by  some  other  member, 
it  is  not  necessary  to  establish  the  intermediate  general  conclu- 


404 


CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

sion  which  affirms  its  possession  by  all  the  members  of  the  class 
in  common.  Now  whether  it  is  so  or  not,  that  principle  of 
order  or  analogy  upon  which  the  reasoning  is  conducted  must 
either  be  stated  or  apprehended  as  a general  truth,  to  give  vali- 
dity to  the  final  conclusion.  In  this  form,  at  least,  the  necessity 
of  general  propositions  as  the  basis  of  inference  is  confirmed, — a 
necessity  which,  however,  I conceive  to  be  involved  in  the  very 
existence,  and  still  more  in  the  peculiar  nature , of  those  faculties 
whose  laws  have  been  investigated  in  this  work.  For  if  the  pro- 
cess of  reasoning  be  carefully  analyzed,  it  will  appear  that  ab- 
straction is  made  of  all  peculiarities  of  the  individual  to  which 
the  conclusion  refers,  and  the  attention  confined  to  those  pro- 
perties by  which  its  membership  of  the  class  is  defined. 

5.  But  besides  the  general  propositions  which  are  derived  by 
induction  from  the  collated  facts  of  experience,  there  exist  others 
belonging  to  the  domain  of  what  is  termed  necessary  truth.  Such 
are  the  general  propositions  of  Arithmetic,  as  well  as  theprojio- 
sitions  expressing  the  laws  of  thought  upon  which  the  general 
methods  of  this  treatise  are  founded;  and  these  propositions 
are  not  only  capable  of  being  rigorously  verified  in  particular 
instances,  but  are  made  manifest  in  all  their  generality  from  the 
study  of  particular  instances.  Again,  there  exist  general  pro- 
positions expressive  of  necessary  truths,  but  incapable,  from  the 
imperfection  of  the  senses,  of  being  exactly  verified.  Some,  if 
not  all,  of  the  propositions  of  Geometry  are  of  this  nature ; but 
it  is  not  in  the  region  of  Geometry  alone  that  such  propositions 
are  found.  The  question  concerning  their  nature  and  origin 
is  a very  ancient  one,  and  as  it  is  more  intimately  connected 
with  the  inquiry  into  the  constitution  of  the  intellect  than  any 
other  to  which  allusion  has  been  made,  it  will  not  be  irrelevant 
to  consider  it  here.  Among  the  opinions  which  have  most 
widely  prevailed  upon  the  subject  are  the  following.  It  has 
been  maintained,  that  propositions  of  the  class  referred  to  exist 
in  the  mind  independently  of  experience,  and  that  those  concep- 
tions which  are  the  subjects  of  them  are  the  imprints  of  eternal 
archetypes.  With  such  archetypes,  conceived,  however,  to  pos- 
sess a reality  of  which  all  the  objects  of  sense  are  but  a faint 
shadow  or  dim  suggestion,  Plato  furnished  his  ideal  world.  It 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT.  405 

has,  on  the  other  hand,  been  variously  contended,  that  the 
subjects  of  such  propositions  are  copies  of  individual  objects  of 
experience ; that  they  are  mere  names ; that  they  are  individual 
objects  of  experience  themselves ; and  that  the  propositions  which 
relate  to  them  are,  on  account  of  the  imperfection  of  those  objects, 
bnt  partially  true;  lastly,  that  they  are  intellectual  products 
formed  by  abstraction  from  the  sensible  perceptions  of  individual 
things,  but  so  formed  as  to  become,  what  the  individual  things 
never  can  be,  subjects  of  science,  i.  e.  subjects  concerning  which 
exact  and  general  propositions  may  be  affirmed.  And  there  ex- 
ist, perhaps,  yet  other  views,  in  some  of  which  the  sensible,  in 
others  the  intellectual  or  ideal,  element  predominates. 

Now  if  the  last  of  the  views  above  adverted  to  be  taken  (for 
it  is  not  proposed  to  consider  either  the  purely  ideal  or  the 
purely  nominalist  view)  and  if  it  be  inquired  what,  in  the 
sense  above  stated,  are  the  proper  objects  of  science,  objects  in 
relation  to  which  its  propositions  are  true  without  any  mixture 
of  error,  it  is  conceived  that  but  one  answer  can  be  given.  It 
is,  that  neither  do  individual  objects  of  experience,  nor  with  all 
probability  do  the  mental  images  which  they  suggest,  possess 
any  strict  claim  to  this  title.  It  seems  to  be  certain,  that  neither 
in  nature  nor  in  art  do  we  meet  with  anything  absolutely  agreeing 
with  the  geometrical  definition  of  a straight  line,  or  of  a triangle, 
or  of  a circle,  though  the  deviation  therefrom  may  be  inappre- 
ciable by  sense ; and  it  may  be  conceived  as  at  least  doubtful, 
whether  we  can  form  a perfect  mental  image,  or  conception,  with 
which  the  agreement  shall  be  more  exact.  But  it  is  not  doubtful 
that  such  conceptions,  however  imperfect,  do  point  to  something 
beyond  themselves,  in  the  gradual  approach  towards  which  all 
imperfection  tends  to  disappear.  Although  the  perfect  triangle, 
or  square,  or  circle,  exists  not  in  nature,  eludes  all  our  powers  of 
representative  conception,  and  is  presented  to  us  in  thought 
only,  as  the  limit  of  an  indefinite  process  of  abstraction,  yet,  by 
a wonderful  faculty  of  the  understanding,  it  may  be  made  the 
subject  of  propositions  which  are  absolutely  true.  The  domain  of 
reason  is  thus  revealed  to  us  as  larger  than  that  of  imagination. 
Should  any,  indeed,  think  that  we  are  able  to  picture  to  ourselves, 
with  rigid  accuracy,  the  scientific  elements  of  form,  direction,  mag- 


406 


CONSTITUTION  OF  THE  INTELLECT.  [CHAF.  XXII. 

nitude,  &c.,  these  things,  as  actually  conceived,  will,  in  the  view 
of  such  persons,  be  the  proper  objects  of  science.  But  if,  as 
seems  to  ine  the  more  just  opinion,  an  incurable  imperfection 
attaches  to  all  our  attempts  to  realize  with  precision  these  ele- 
ments, then  we  can  only  affirm,  that  the  more  external  objects 
do  approach  in  reality,  or  the  conceptions  of  fancy  by  abstraction, 
to  certain  limiting  states,  never,  it  may  be,  actually  attained,  the 
more  do  the  general  propositions  of  science  concerning  those 
things  or  conceptions  approach  to  absolute  truth,  the  actual  devi- 
ation therefrom  tending  to  disappear.  To  some  extent,  the  same 
observations  are  applicable  also  to  the  physical  sciences.  What 
have  been  termed  the  “fundamental  ideas”  of  those  sciences  as 
force,  polarity,  crystallization,  &c.,*  are  neither,  as  I conceive, 
intellectual  products  independent  of  experience,  nor  mere  copies 
of  external  things ; but  while,  on  the  one  hand,  they  have  a ne- 
cessary antecedent  in  experience,  on  the  other  hand  they  require 
for  their  formation  the  exercise  of  the  power  of  abstraction,  in 
obedience  to  some  general  faculty  or  disposition  of  our  nature, 
which  ever  prompts  us  to  the  research,  and  qualifies  us  for  the 
appreciation,  of  order. f Thus  we  study  approximately  the  effects 
of  gravitation  on  the  motions  of  the  heavenly  bodies,  by  a re- 
ference to  the  limiting  supposition,  that  the  planets  are  perfect 


* Whe well’s  Philosophy  of  the  Inductive  Sciences,  pp.  7b  77,  ‘213. 
f Of  the  idea  of  order  it  has  been  profoundly  said,  that  it  carries  within  itself 
its  own  justification  or  its  own  control,  the  very  trustworthiness  of  our  faculties 
being  judged  by  the  conformity  of  their  results  to  an  order  which  satisfies  the 
reason.  “ L’idee  de  l’ordre  a cela  de  singulier  et  d’eminent,  qu’elle  porte  en  elle 
meme  sa  justification  ou  son  controle.  Pour  trouver  si  nos  autres  faculty  nous 
trompent  ou  nous  ne  trompent  pas,  nous  examinons  si  les  notions  qu'elles  nous 
donnent  s’enchainent  on  ne  s’enchainent  pas  suivant  un  ordre  qui  satisfasse  la 
raison.” — Cournot,  Essai  sur  les  fondements  de  nos  Connaissanees.  Admitting  this 
principle  as  the  guide  of  those  powers  of  abstraction  which  we  undoubtedly  pos- 
sess, it  seems  unphilosophical  to  assume  that  the  fundamental  ideas  of  the 
sciences  are  not  derivable  from  experience.  Doubtless  the  capacities  which 
have  been  given  to  us  for  the  comprehension  of  the  actual  world  would  avail  us 
in  a differently  constituted  scene,  if  in  some  form  or  other  the  dominion  of 
order  was  still  maintained.  It  is  conceivable  that  in  such  a new  theatre  of  spe- 
culation, the  laws  of  the  intellectual  procedure  remaining  the  same,  the  funda- 
mental ideas  of  the  sciences  might  be  wholly  different  from  those  with  which  we 
are  at  present  acquainted. 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT.  407 

spheres  or  spheroids.  We  determine  approximately  the  path 
of  a ray  of  light  through  the  atmosphere,  by  a process  in  which 
abstraction  is  made  of  all  disturbing  influences  of  temperature. 
And  such  is  the  order  of  procedure  in  all  the  higher  walks  of 
human  knowledge.  Now  what  is  remarkable  in  connexion  with 
these  processes  of  the  intellect  is  the  disposition,  and  the  cor- 
responding ability,  to  ascend  from  the  imperfect  representations 
of  sense  and  the  diversities  of  individual  experience,  to  the  per- 
ception of  general,  and  it  may  be  of  immutable  truths.  Where- 
ever  this  disposition  and  this  ability  unite,  each  series  of  con- 
nected facts  in  nature  may  furnish  the  intimations  of  an  order 
more  exact  than  that  which  it  directly  manifests.  For  it  may 
serve  as  ground  and  occasion  for  the  exercise  of  those  powers, 
whose  office  it  is  to  apprehend  the  general  truths  which  are  in- 
deed exemplified,  but  never  with  perfect  fidelity,  in  a world  of 
changeful  phenomena. 

6.  The  truth  that  the  ultimate  laws  of  thought  are  mathe- 
matical in  their  form,  viewed  in  connexion  with  the  fact  of  the 
possibility  of  error,  establishes  a ground  for  some  remarkable  con- 
clusions. If  we  directed  our  attention  to  the  scientific  truth 
alone,  we  might  be  led  to  infer  an  almost  exact  parallelism  be- 
tween the  intellectual  operations  and  the  movements  of  external 
nature.  Suppose  any  one  conversant  with  physical  science,  but 
unaccustomed  to  reflect  upon  the  nature  of  his  own  faculties,  to 
have  been  informed,  that  it  had  been  proved,  that  the  laws  of 
those  faculties  were  mathematical ; it  is  probable  that  after  the 
first  feelings  of  incredulity  had  subsided,  the  impression  would 
arise,  that  the  order  of  thought  must,  therefore,  be  as  neces- 
sary as  that  of  the  material  universe.  We  know  that  in  the 
realm  of  natural  science,  the  absolute  connexion  between  the 
initial  and  final  elements  of  a problem,  exhibited  in  the  mathe- 
matical form,  fitly  symbolizes  that  physical  necessity  which  binds 
together  effect  and  cause.  The  necessary  sequence  of  states  and 
conditions  in  the  inorganic  world,  and  the  necessary  connexion 
of  premises  and  conclusion  in  the  processes  of  exact  demonstra- 
tion thereto  applied,  seem  to  be  co-ordinate.  It  may  possibly  be 
a question,  to  which  of  the  two  series  the  primary  application  of 
the  term  “necessary”  is  due;  whether  to  the  observed  constancy  of 


408 


CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

Nature,  or  to  the  indissoluble  connexion  of  propositions  in  all  valid 
reasoning  upon  her  works.  Historically  we  should  perhaps  give 
the  preference  to  the  former,  philosophically  to  the  latter  view. 
But  the  fact  of  the  connexion  is  indisputable,  and  the  analogy  to 
which  it  points  is  obvious. 

Were,  then,  the  laws  of  valid  reasoning  uniformly  obeyed,  a 
very  close  parallelism  would  exist  between  the  operations  of  the 
intellect  and  those  of  external  Nature.  Subjection  to  laws  ma- 
thematical in  their  form  and  expression,  even  the  subjection  of 
an  absolute  obedience,  would  stamp  upon  the  two  series  one 
common  character.  The  reign  of  necessity  over  the  intellectual 
and  the  physical  world  would  be  alike  complete  and  universal. 

But  while  the  observation  of  external  Nature  testifies  with 
ever-strengthening  evidence  to  the  fact,  that  uniformity  of 
operation  and  unvarying  obedience  to  appointed  laws  prevail 
throughout  her  entire  domain,  the  slightest  attention  to  the  pro- 
cesses of  the  intellectual  world  reveals  to  us  another  state  of 
things.  The  mathematical  laws  of  reasoning  are,  properly  speak- 
ing, the  laws  of  right  reasoning  only,  and  their  actual  transgres- 
sion is  a perpetually  recurring  phamomenon-.  Error,  which  has 
no  place  in  the  material  system,  occupies  a large  one  here.  We 
must  accept  this  as  one  of  those  ultimate  facts,  the  origin  of  which 
it  lies  beyond  the  province  of  science  to  determine.  We  must 
admit  that  there  exist  laws  which  even  the  rigour  of  their  ma- 
thematical forms  does  not  preserve  from  violation.  We  must 
ascribe  to  them  an  authority  the  essence  of  which  does  not  con- 
sist in  power,  a supremacy  which  the  analogy  of  the  inviolable 
order  of  the  natural  world  in  no  way  assists  us  to  comprehend. 

As  the  distinction  thus  pointed  out  is  real,  it  remains  un- 
affected by  any  peculiarity  in  our  views  respecting  other  portions 
of  the  mental  constitution.  If  Ave  regard  the  intellect  as  free, 
and  this  is  apparently  the  view  most  in  accordance  Avith  the  gene- 
ral spirit  of  these  speculations,  its  freedom  must  be  vieAved  as 
opposed  to  the  dominion  of  necessity,  not  to  the  existence  of  a 
certain  just  supremacy  of  truth.  The  laws  of  correct  inference 
may  be  violated,  but  they  do  not  the  less  truly  exist  on  this  ac- 
count. Equally  do  they  remain  unaffected  in  character  and  au- 
thority if  the  hypothesis  of  necessity  in  its  extreme  form  be 


CHAP.  XXII.]  . CONSTITUTION  OF  THE  INTELLECT. 


409 


adopted.  Let  it  be  granted  that  the  laws  of  valid  reasoning, 
such  as  they  are  determined  to  be  in  this  work,  or,  to  speak  more 
generally,  such  as  they  would  finally  appear  in  the  conclusions  of 
an  exhaustive  analysis,  form  but  a part  of  the  system  of  laws  by 
which  the  actual  processes  of  reasoning,  whether  right  or  wrong, 
are  governed.  Let  it  be  granted  that  if  that  system  were  known 
to  us  in  its  completeness,  we  should  perceive  that  the  whole  in- 
tellectual procedure  was  necessary,  even  as  the  movements  of  the 
inorganic  world  are  necessary.  And  let  it  finally,  as  a conse- 
quence of  this  hypothesis,  be  granted  that  the  phenomena  of  in- 
correct reasoning  or  error,  wheresoever  presented,  are  due  to  the 
interference  of  other  laws  with  those  laws  of  which  right  reason- 
ing is  the  product.  Still  it  would  remain  that  there  exist  among 
the  intellectual  laws  a number  marked  out  from  the  rest  by  this 
special  character,  viz.,  that  every  movement  of  the  intellectual 
system  which  is  accomplished  solely  under  their  direction  is 
right,  that  every  interference  therewith  by  other  laws  is  not  in- 
terference only,  but  violation.  It  cannot  but  be  felt  that  this 
circumstance  would  give  to  the  laws  in  question  a character  of 
distinction  and  of  predominance.  They  would  but  the  more 
evidently  seem  to  indicate  a final  purpose  which  is  not  always 
fulfilled,  to  possess  an  authority  inherent  and  just,  but  not 
always  commanding  obedience. 

Now  a little  consideration  will  show  that  there  is  nothing: 
analogous  to  this  in  the  government  of  the  world  by  natural  law. 
The  realm  of  inorganic  Nature  admits  neither  of  preference  nor 
of  distinctions.  We  cannot  separate  any  portion  of  her  laws 
from  the  rest,  and  pronounce  them  alone  worthy  of  obedience, — 
alone  charged  with  the  fulfilment  of  her  highest  purpose.  On 
the  contrary,  all  her  laws  seem  to  stand  co-ordinate,  and  the 
larger  our  acquaintance  with  them,  the  more  necessary  does  their 
united  action  seem  to  the  harmony  and,  so  far  as  we  can  com- 
prehend it,  to  the  general  design  of  the  system.  How  often  the 
most  signal  departures  from  apparent  order  in  the  inorganic 
world,  such  as  the  perturbations  of  the  planetary  system,  the  in- 
terruption of  the  process  of  crystallization  by  the  intrusion  of  a 
foreign  force,  and  others  of  a like  nature,  either  merge  into  the 
conception  of  some  more  exalted  scheme  of  order,  or  lose  to  a 


410 


CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

more  attentive  and  instructed  gaze  their  abnormal  aspect,  it  is 
needless  to  remark.  One  explanation  only  of  these  facts  can  be 
given,  viz.,  that  the  distinction  between  true  and/o/se,  between 
correct  and  incorrect , exists  in  the  processes  of  the  intellect,  but 
not  in  the  region  of  a physical  necessity.  As  we  advance  from 
the  lower  stages  of  organic  being  to  the  higher  grade  of  conscious 
intelligence,  this  contrast  gradually  dawns  upon  us.  Wherever 
the  phenomena  of  life  are  manifested,  the  dominion  of  rigid  law 
in  some  degree  yields  to  that  mysterious  principle  of  activity. 
Thus,  although  the  structure  of  the  animal  tribes  is  conformable 
to  certain  general  types,  yet  are  those  types  sometimes,  perhaps, 
in  relation  to  the  highest  standards  of  beauty  and  proportion, 
always,  imperfectly  realized.  The  two  alternatives,  between 
which  Art  in  the  present  day  fluctuates,  are  the  exact  imitation 
of  individual  forms,  and  the  endeavour,  by  abstraction  from  all 
such,  to  arrive  at  the  conception  of  an  ideal  grace  and  expression, 
never,  it  may  be,  perfectly  manifested  in  forms  of  earthly  mould. 
Again,  those  teleological  adaptations  by  which,  without  the  or- 
ganic type  being  sacrificed,  species  become  fitted  to  new  con- 
ditions or  abodes,  are  but  slowly  accomplished, — accomplished, 
however,  not,  apparently,  by  the  fateful  power  of  external  cir- 
cumstances, but  by  the  calling  forth  of  an  energy  from  within. 
Life  in  all  its  forms  may  thus  be  contrasted  with  the  passive  fixity 
of  inorganic  nature.  But  inasmuch  as  the  perfection  of  the  types 
in  which  it  is  corporeally  manifested  is  in  some  measure  of  an 
ideal  character,  inasmuch  as  we  cannot  precisely  define  the 
highest  suggested  excellency  of  form  and  of  adaptation,  the  con- 
trast is  less  marked  here  than  that  which  exists  between  the  in- 
tellectual processes  and  those  of  the  purely  material  world.  For 
the  definite  and  technical  character  of  the  mathematical  laws  by 
which  both  are  governed,  places  in  stronger  light  the  fundamental 
difference  between  the  kind  of  authority  which,  in  their  capacity 
of  government,  they  respectively  exercise. 

7.  There  is  yet  another  instance  connected  with  the  general 
objects  of  this  chapter,  in  which  the  collation  of  truths  or  facts, 
drawn  from  different  sources,  suggests  an  instructive  train  of  re- 
flection. It  consists  in  the  comparison  of  the  laws  of  thought,  in 
their  scientific  expression,  with  the  actual  forms  which  physical 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT. 


411 


speculation  in  early  ages,  and  metaphysical  speculation  in  all 
ages,  have  tended  to  assume.  There  are  two  illustrations  of  this 
remark,  to  which,  in  particular,  I wish  to  direct  attention  here. 

1st.  It  has  been  shown  (III.  13)  that  there  is  a scientific 
connexion  between  the  conceptions  of  unity  in  Number,  and  the 
universe  in  Logic.  They  occupy  in  their  respective  systems  the 
same  relative  place,  and  are  subject  to  the  same  formal  laws. 
Now  to  the  Greek  mind,  in  that  early  stage  of  activity, — a stage 
not  less  marked,  perhaps  not  less  necessary,  in  the  progression  of 
the  human  intellect,  than  the  era  of  Bacon  or  of  Newton,  — when 
the  great  problems  of  Nature  began  to  unfold  themselves,  while 
the  means  of  observation  were  as  yet  wanting,  and  its  necessity 
not  understood,  the  terms  “ Universe”  and  “ The  One”  seem  to 
have  been  regarded  as  almost  identical.  To  assign  the  nature  of 
that  unity  of  which  all  existence  was  thought  to  be  a manifesta- 
tion, was  the  first  aim  of  philosophy.*  Thales  sought  for  this 
fundamental  unity  in  water.  Anaximenes  and  Diogenes  con- 
ceived it  to  be  air.  Hippasus  of  Metapontum,  and  Heraclitus 
the  Ephesian,  pronounced  that  it  was  fire.  Less  definite  or 
less  confident  in  his  views,  Parmenides  simply  declared  that  all 
existing  things  were  One;  Melissus  that  the  Universe  was  infi- 
nite, unsusceptible  of  change  or  motion,  One,  like  to  itself,  and 
that  motion  was  not,  but  seemed  to  be.f  In  a spirit  which,  to  the 
reflective  mind  of  Aristotle,  appeared  sober  when  contrasted 
with  the  rashness  of  previous  speculation,  Anaxagoras  of  Clazo- 
mente,  following,  perhaps,  the  steps  of  his  fellow-citizen,  Hermo- 
timus,  sought  in  Intelligence  the  cause  of  the  world  and  of  its 
order. | The  pantheistic  tendency  which  pervaded  many  of  these 
speculations  is  manifest  in  the  language  of  Xenophanes,  the 
founder  of  the  Eleatic  school,  who,  “ surveying  the  expanse  of 


• See  various  passages  in  Aristotle’s  Metaphysics,  Book  i. 

■f  ’E^oicfi  Si  avTif  to  xdv  axtipov  tlvai,  sat  avaWoiuiTov,  ical  o.k'ivt)Tov,  > :ai 
tv,  opioiov  iavTip  icai  xXijptg.  Kivgaiv  Tt  pi)  tlvai  SoKtiv  St  tlvai. — Diog.  Laert.  IX. 
cap.  4. 

J Not/v  Sr]  rig  tixujv  IvtTvai , xadaxtp  iv  rolg  Zipoig,  xai  tv  rg  <pl)<rti,  rbv 
a’lTiov  Toil  Koapov  icat  rfjg  ra^tiog  xdatjg  olov  vrjifxov  iipavr)  Trap’  ting  \iyovrag 
Tovg  xpOTtpov.  $avtpH>g  fitv  ovv  ’A vaZayopav  iaptv  atpapitvov  tovtivv  twv  \o- 
yiov,  aiTiav  S'  l\ti  xportpov  'EppoTipog  6 KXaZo/itviog  tixtiv. — Arist.  Met.  I.  3. 


412 


CONSTITUTION  OF  THE  INTELLECT.  [cHAP.  XXII. 

heaven,  declared  that  the  One  was  God.”*  Perhaps  there  are  few, 
If  any,  of  the  forms  in  which  unity  can  be  conceived,  in  the  ab- 
stract as  numerical  or  rational,  in  the  concrete  as  a passive  sub- 
stance, or  a central  and  living  principle,  of  which  we  do  not 
meet  with  applications  in  these  ancient  doctrines.  The  writings 
of  Aristotle,  to  which  I have  chiefly  referred,  abound  with  allu- 
sions of  this  nature,  though  of  the  larger  number  of  those  who 
once  addicted  themselves  to  such  speculations,  it  is  probable  that 
the  very  names  have  perished.  Strange,  but  suggestive  truth, 
that  while  Nature  in  all  but  the  aspect  of  the  heavens  must  have 
appeared  as  little  else  than  a scene  of  unexplained  disorder,  while 
the  popular  belief  was  distracted  amid  the  multiplicity  of  its  gods, 
— the  conception  of  a primal  unity,  if  only  in  a rude,  material  form, 
should  have  struck  deepest  root ; surviving  in  many  a thought- 
ful breast  the  chills  of  a lifelong  disappointment,  and  an  endless 
search  !f 

2ndly.  In  equally  intimate  alliance  with  that  law  of  thought 
which  is  expressed  by  an  equation  of  the  second  degree,  and 
which  has  been  termed  in  this  treatise  the  law  of  duality,  stands 
the  tendency  of  ancient  thought  to  those  forms  of  philosophical 
speculation  which  are  known  under  the  name  of  dualism.  The 
theory  of  Empedocles, | which  explained  the  apparent  contradic- 
tions of  nature  by  referring  them  to  the  two  opposing  principles 


* Etvo<pav7)g  Si . . . tig  tov  bXov  ovpavov  axofSX'tipag,  to  tv  tivai  <pr)Oi  t'ov 
Qtov. — lb. 

f The  following  lines,  preserved  by  Sextus  Empiricus,  and  ascribed  to  Timon 
the  Sillograph,  are  not  devoid  of  pathos  : — 

otg  ical  eyibv  bipiXov  xvkivov  voov  avTi(io\ri<mi 
aptyoTtpojiXtXTog  (SoXiy  S’  boip  tXtxaTtidrjV, 
xptofivytvfig  tr  iibv)  Kai  avafMpripiGTog  axaopg 
GKtnToavvpQ’  oxTrt]  yap  ipbv  voov  tipvaaipi, 
tig  iv  r’  avro  rt  xav  aviXvtTO. 

I quote  them  from  Ritter,  and  venture  to  give  the  following  version: — 

Be  mine,  to  partial  views  no  more  confin’d 
Or  sceptic  doubts,  the  truth-illumin’d  mind  ! 

For,  long  deceiv’d,  yet  still  on  Truth  intent, 

Life’s  waning  years  in  wand’rings  wild  are  spent. 

Still  restless  thought  the  same  high  quest  essays, 

And  still  the  One,  the  All,  eludes  my  gaze. 

J Arist.  Met.  t.  4.  6. 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT. 


413 


of  “strife”  and  “ friendship and  the  theory  of  Leucippus,* * * § 
which  resolved  all  existence  into  the  two  elements  of  a plenum 
and  a vacuum,  are  of  this  nature.  The  famous  comparison  of  the 
universe  to  a lyre  or  a bow,t  its  “recurrent  harmony”  being  the 
product  of  opposite  states  of  tension,  betrays  the  same  origin. 
In  the  system  of  Pythagoras,  which  seems  to  have  been  a combi- 
nation of  dualism  with  other  elements  derived  from  the  study  of 
numbers,  and  of  their  relations,  ten  fundamental  antitheses  are 
recognised:  finite  and  infinite,  even  and  odd,  unity  and  multitude, 
right  and  left,  male  and  female,  rest  and  motion,  straight  and 
curved,  light  and  darkness,  good  and  evil,  the  square  and  the 
oblong.  In  that  of  Alcmaeon  the  same  fundamental  dualism  is 
accepted,  but  without  the  definite  and  numerical  limitation  with 
which  it  is  connected  in  the  Pythagorean  system.  The  grand 
development  of  this  idea  is,  however,  met  with  in  that  ancient 
Manichaean  doctrine,  which  not  only  formed  the  basis  of  the  re- 
ligious system  of  Persia,  but  spread  widely  through  other  regions 
of  the  East,  and  became  memorable  in  the  history  of  the  Christian 
Church.  The  origin  of  dualism  as  a speculative  opinion,  not 
yet  connected  with  the  personification  of  the  Evil  Principle,  but 
naturally  succeeding  those  doctrines  which  had  assumed  the 
primal  unity  of  Nature,  is  thus  stated  by  Aristotle : — “ Since 
there  manifestly  existed  in  Nature  things  opposite  to  the  good, 
and  not  only  order  and  beauty,  but  also  disorder  and  deformity  ; 
and  since  the  evil  things  did  manifestly  preponderate  in  number 
over  the  good,  and  the  deformed  over  the  beautiful,  some  one 
else  at  length  introduced  strife  and  friendship  as  the  respective 
causes  of  these  diverse  phienomena.”J  And  in  Greece,  indeed, 
it  seems  to  have  been  chiefly  as  a philosophical  opinion,  or  as  an 
adjunct  to  philosophical  speculation,  that  the  dualistic  theory  ob- 
tained ground. § The  moral  application  of  the  doctrine  most  in 


* Arist.  Met.  i.  4,  9. 

f TraXivrpoirog  appovit]  OKoig  Trip  roZov  Kai  Xvpyg Heraclitus,  quoted  in 

Oriyenis  Philosophumena,  IX.  9.  Also  Plutarch,  He  Iside  et  Osiride. 

J ’Etti'i  8i  <ai  ravavria  roTg  ayaGolg  Ivovra  i<j>aivero  iv  rj?  tpvtrn,  Kai  ov 
pbvov  raZig  Kai  to  KaXov  aXXa  Kai  araZia  Kai  to  aiaxpov,  Kai  TrXiito  ra  KaKU 
tuiv  ayaduiv  Kai  ra  tpavXa  roi v KaXibv,  ovrmg  uXXog  rig  ipiXiav  lioyviyKt  Kai  vei- 
Kog,  iKartpov  iKaripuiu  ainov  tovtujv. — Arist.  Metaphysica,  I.  4. 

§ Witness  Aristotle’s  well-known  derivation  of  the  cdements  from  the  quali- 


414  CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

accordance  with  the  Greek  mind  is  preserved  in  the  great  Pla- 
tonic antithesis  of  “ being  and  non-being,” — the  connexion  of  the 
former  w ith  whatsoever  is  good  and  true,  with  the  eternal  ideas, 
and  the  archetypal  world : of  the  latter  with  evil,  with  error, 
with  the  perishable  phenomena  of  the  present  scene.  The  two 
forms  of  speculation  which  we  have  considered  were  here  blended 
together ; nor  was  it  during  the  youth  and  maturity  of  Greek 
philosophy  alone  that  the  tendencies  of  thought  above  described 
were  manifested.  Ages  of  imitation  caught  up  and  adopted  as 
their  own  the  same  spirit.  Especially  wherever  the  genius  of 
Plato  exercised  sway  was  this  influence  felt.  The  unity  of  all 
real  being,  its  identity  with  truth  and  goodness  considered 
as  to  their  essence  ; the  illusion,  the  profound  unreality,  of  all 
merely  phaenomenal  existence ; such  were  the  views, — such  the 
dispositions  of  thought,  which  it  chiefly  tended  to  foster.  Hence 
that  strong  tendency  to  mysticism  which,  when  the  days  of  re- 
nown, whether  on  the  field  of  intellectual  or  on  that  of  social  en- 
terprise, had  ended  in  Greece,  became  prevalent  in  her  schools 
of  philosophy,  and  reached  their  culminating  point  among  the 
Alexandrian  Platonists.  The  supposititious  treatises  of  Dionysius 
the  Areopagite  served  to  convey  the  same  influence,  much  modi- 
fied by  its  contact  with  Aristotelian  doctrines,  to  the  scholastic 
disputants  of  the  middle  ages.  It  can  furnish  no  just  ground  of 
controversy  to  say,  that  the  tone  of  thought  thus  encouraged  was 
as  little  consistent  with  genuine  devotion  as  with  a sober  phi- 
losophy. That  kindly  influence  of  human  affections,  that  homely 
intercourse  with  the  common  things  of  life,  which  form  so  large 
a part  of  the  true,  because  intended,  discipline  of  our  nature, 
would  be  ill  replaced  by  the  contemplation  even  of  the  highest 
object  of  thought,  viewed  by  an  excessive  abstraction  as  some- 
thing concerning  which  not  a single  intelligible  proposition  could 
either  be  affirmed  or  denied.* *  I would  but  slightly  allude  to 
those  connected  speculations  on  the  Divine  Nature  which  ascribed 


ties  “ warm,”  and  “ dry,”  and  their  contraries.  It  is  characteristic  that  Plato 
connects  their  generation  with  mathematical  principles. — Timaus,  cap.  xi. 

• A vroc  Kai  inrip  Q'tcnv  Iitti  Kai  a<paipiaiv. — Dion.  Areop.  De  Divinis  No- 
minibus, cap.  II. 


415 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT. 

to  it  the  perfect  union  of  opposite  qualities,* * * §  or  to  the  remarkable 
treatises  of  Anselm,  designed  to  establish  a theory  of  tbe  universe 
upon  the  analogies  of  thought  and  being. f The  primal  unity  is 
there  represented  as  having  its  abode  in  the  one  eternal  Truth. 
The  conformity  of  Nature  to  her  laws,  the  obedience  of  moral 
agents  to  the  dictates  of  rectitude,  are  the  same  Truth  seen  in 
action;  the  world  itself  being  but  an  expression  of  the  self-reflect- 
ing thought  of  its  Author.}  Still  more  marked  was  the  revival 
of  the  older  forms  of  speculation  during  the  sixteenth  and  seven- 
teenth centuries.  The  friends  and  associates  of  Lorenzo  the 
Magnificent,  the  recluses  known  in  England  as  the  Cambridge 
Platonists,  together  with  many  meditative  spirits  scattered 
through  Europe,  devoted  themselves  anew,  either  to  the  task  of 
solving  the  ancient  problem,  De  Uno,  Vero,  Bono,  or  to  that  of 
proving  that  all  such  inquiries  are  futile  and  vain.  § The  logical 
elements  which  underlie  all  these  speculations,  and  from  which 
they  appear  to  borrow  at  least  their  form,  it  would  be  easy  to 
trace  in  the  outlines  of  more  modern  systems, — more  especially 
in  that  association  of  the  doctrine  of  the  absolute  unity  with  the 
distinction  of  the  ego  and  the  non-ego  as  the  type  of  Nature, 
which  forms  the  basis  of  the  philosophy  of  Hegel.  The  attempts 
of  speculative  minds  to  ascend  to  some  high  pinnacle  of  truth, 
from  which  they  might  survey  the  entire  framework  and  con- 


• See  especially  the  lofty  strain  of  Hildebert  beginning  “ Alpha  et  Q magno 
Deus.”  (Trench’s  Sacred  Latin  Poetry.)  The  principle  upon  which  all  these 
speculations  rest  is  thus  stated  in  the  treatise  referred  to  in  the  last  note. 
OiiS'tv  ovv  aronov,  apvbpiiv  iucovwv  ini  to  navroiv  ainov  avafiavrag,  bmp- 
Koopioig  o<p6a\polg  Qnopi]aai  navra  iv  t tp  navriov  airily,  Kai  Ta  a\\T)\oig  ivav- 

ria  povouSCjg  Kai  yjvojptviog De  Diviirs  Novninibus , cap.  v.  • And  the  kind  of 

knowledge  which  it  is  thus  sought  to  attain  is  described  as  a “ darkness  beyond 
light,”  vmp<pCiTog  yvo<pog.  ( De  Mystica  Theologia,  cap.  I.)  Milton  has  a simi- 
lar thought — 

“ Dark  with  excessive  bright  Thy  skirts  appear.” 

Par.  Lost,  Book  III. 

Contrast  with  these  the  nobler  simplicity  of  1 John,  i.  5. 

f Monologium,  Prosologium,  and  De  Veritate. 

J “ Idcirco  cum  ipse  summus  spiritus  dicit  seipsum  dicit  omnia  quae  facta 
sunt.” — Monolog,  cap.  xxm. 

§ See  dissertations  in  Spinoza,  Picus  of  Mirandula,  H.  More,  &c.  Modern 
discussions  of  this  nature  are  chiefly  in  connexion  with  aesthetics,  the  ground  of 
the  application  being  contained  in  the  formula  of  Augustine  : “ Omnis  porro 
pulchritudinis  forma,  unitas  est.” 


416 


CONSTITUTION  OF  THE  INTELLECT.  [cHAP.  XXII. 

nexion  of  things  in  the  order  of  deductive  thought , have  differed 
less  in  the  forms  of  theory  which  they  have  produced,  than 
through  the  nature  of  the  interpretations  which  have  been  as- 
signed to  those  forms.*  And  herein  lies  the  real  question  as  to 
the  influence  of  philosophical  systems  upon  the  disposition  and 
the  life.  For  though  it  is  of  slight  moment  that  men  should 
agree  in  tracing  back  all  the  forms  and  conditions  of  being  to  a 
primal  unity,  it  is  otherwise  as  concerns  their  conceptions  of 
what  that  unity  is,  and  what  are  the  kinds  of  relation,  beside 
that  of  mere  causality,  which  it  sustains  to  themselves.  Herein 
too  may  be  felt  the  powerlessness  of  mere  Logic,  the  insufficiency 
of  the  profoundest  knowledge  of  the  laws  of  the  understanding, 
to  resolve  those  problems  which  lie  nearer  to  our  hearts,  as  pro- 
gressive years  strip  away  from  our  life  the  illusions  of  its  golden 
dawn. 

8.  If  the  extremely  arbitrary  character  of  human  opinion  be 
considered,  it  will  not  be  expected,  nor  is  it  here  maintained,  that 
the  above  are  the  only  forms  in  which  speculative  men  have 
shaped  their  conjectural  solutions  of  the  problem  of  existence. 
Under  particular  influences  other  forms  of  doctrine  have  arisen, 
not  unfrequently,  however,  masking  those  portrayed  above.f 


* For  instance,  the  learned  mysticism  of  Gioberti,  widely  as  it  differs  in  its 
spirit  and  its  conclusions  from  the  pantheism  of  Hegel  (both  being,  perhaps, 
equally  remote  from  truth),  resembles  it  in  applying  both  to  thought  and 
to  being  the  principles  of  unity  and  duality.  It  is  asked  Or  non  e egli 
chiaro  che  ogni  discorso  si  riduce  in  fine  in  fine  alle  idee  di  Dio,  del  raondo,  e 
della  creazione,  l’ultima  delle  quali  e il  legame  delle  due  prime  ?”  And  this  ques- 
tion being  affirmatively  answered  in  the  formula,  “ l'Ente  crea  le  esistenze,”  it 
is  said  of  that  formula,  — “ Essa  abbraccia  la  realta  universale  nella  dualita  del 
necessario  e del  contingente,  esprime  il  vincolo  di  questi  due  ordini,  e collocan- 
dolo  nella  creazion  sostanziale,  riduce  la  dualita  reale  a un  principio  unico,  all 
unit&  primordiale  dell’  Ente  non  astratto,  complessivo,  e generico,  ma  concreto, 
individuato,  assoluto,  e creatore.” — Del  Bello  e del  Buono,  pp.  30,  31. 

f Evidence  in  support  of  this  statement  will  be  found  in  the  remarkable 
treatise  recently  published  under  the  title  (the  correctness  of  which  seems  doubt- 
ful) of  Origenis  Philosophumena.  The  early  corruptions  of  Christianity  of  which 
it  contains  the  record,  though  many  of  them,  as  is  evident  from  their  Ophite 
character,  derived  from  the  very  dregs  of  paganism,  manifest  certain  persistent 
forms  of  philosophical  speculation.  For  the  most  part  they  either  belong  to  the 
dualistic  scheme,  or  recognise  three  principles,  primary  or  derived,  between  two 
of  which  the  dualistic  relation  may  be  traced — Orig.  Phil.,  pp.  135,  139,  150, 
235,  253,  204. 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT.  417 

But  the  wide  prevalence  of  the  particular  theories  which  we  have 
considered,  together  with  their  manifest  analogy  with  the  ex- 
pressed laws  of  thought,  may  justly  be  conceived  to  indicate  a 
connexion  between  the  two  systems.  As  all  other  mental  acts 
and  procedures  are  beset  by  their  peculiar  fallacies,  so  the  opera- 
tion of  that  law  of  thought  termed  in  this  work  the  law  of  duality 
may  have  its  own  peculiar  tendency  to  error,  exalting  mere  want 
of  agreement  into  contrariety,  and  thus  form  a world  which  we 
necessarily  view  as  formed  of  parts  supplemental  to  each  other, 
framing  the  conception  of  a world  fundamentally  divided  by  op- 
posing powers.  Such,  with  some  large  but  hasty  inductions  from 
phenomena,  may  have  been  the  origin  of  dualism, — indepen- 
dently of  the  question  whether  dualism  is  in  any  form  a true 
theory  or  not.  Here,  however,  it  is  of  more  importance  to  con- 
sider in  detail  the  bearing  of  these  ancient  forms  of  speculation, 
as  revived  in  the  present  day,  upon  the  progress  of  real  know- 
ledge ; and  upon  this  point  I desire,  in  pursuance  of  what  has 
been  said  in  the  previous  section,  to  add  the  following  remarks  : 
1st.  All  sound  philosophy  gives  its  verdict  against  such  spe- 
culations, if  regarded  as  a means  of  determining  the  actual  con- 
stitution of  things.  It  may  be  that  the  progress  of  natural 
knowledge  tends  towards  the  recognition  of  some  central  Unity 
in  Nature.  Of  such  unity  as  consists  in  the  mutual  relation  of 
the  parts  of  a system  there  can  be  little  doubt,  and  able  men 
have  speculated,  not  without  grounds,  on  a more  intimate  corre- 
lation of  physical  forces  than  the  mere  idea  of  a system  would 
lead  us  to  conjecture.  Further,  it  may  be  that  in  the  bosom  of 
that  supposed  unity  are  involved  some  general  principles  of  di- 
vision and  re-union,  the  sources,  under  the  Supreme  Will,  of  much 
of  the  related  variety  of  Nature.  The  instances  of  sex  and  po- 
larity have  been  adduced  in  support  of  such  a view.  As  a sup- 
position, I will  venture  to  add,  that  it  is  not  very  improbable 
that,  in  some  such  way  as  this,  the  constitution  of  things  without 
may  correspond  to  that  of  the  mind  within.  But  such  corres- 
pondence, if  it  shall  ever  be  proved  to  exist,  will  appear  as  the 
last  induction  from  human  knowledge,  not  as  the  first  principle 
of  scientific  inquiry.  The  natural  order  of  discovery  is  from  the 
particular  to  the  universal,  and  it  may  confidently  be  affirmed 


418  CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

that  we  have  not  yet  advanced  sufficiently  far  on  this  track  to 
enable  us  to  determine  what  are  the  ultimate  forms  into  which  all 
the  special  differences  of  Nature  shall  merge,  and  from  which 
they  shall  receive  their  explanation. 

2ndly.  Were  this  correspondence  between  the  forms  of  thought 
and  the  actual  constitution  of  Nature  proved  to  exist,  whatso- 
ever connexion  or  relation  it  might  be  supposed  to  establish  be- 
tween the  two  systems,  it  would  in  no  degree  affect  the  question 
of  their  mutual  independence.  It  would  in  no  sense  lead  to  the 
consequence  that  the  one  system  is  the  mere  product  of  the  other. 
A too  great  addiction  to  metaphysical  speculations  seems,  in 
some  instances,  to  have  produced  a tendency  toward  this  species 
of  illusion.  Thus,  among  the  many  attempts  which  have  been 
made  to  explain  the  existence  of  evil,  it  has  been  sought  to  assign 
to  the  fact  a merely  relative  character, — to  found  it  upon  a species 
of  logical  opposition  to  the  equally  relative  element  of  good.  It 
suffices  to  say,  that  the  assumption  is  purely  gratuitous.  What 
evil  may  be  in  the  eyes  of  Infinite  wisdom  and  purity,  we  ean  at 
the  best  but  dimly  conjecture;  but  to  us,  in  all  its  forms,  whe- 
ther of  pain  or  defect,  or  moral  transgression,  or  retributory  wo, 
it  can  wear  but  one  aspect, — that  of  a sad  and  stern  reality, 
against  which,  upon  somewhat  more  than  the  highest  order  of 
prudential  considerations,  the  whole  preventive  force  of  our 
nature  may  be  exerted.  Now  what  has  been  said  upon  the 
particular  question  just  considered,  is  equally  applicable  to  many 
other  of  the  debated  points  of  philosophy ; such,  for  instance, 
as  the  external  reality  of  space  and  time.  We  have  no  war- 
rant for  resolving  these  into  mere  forms  of  the  understanding, 
though  they  unquestionably  determine  the  present  sphere  of 
our  knowledge.  And,  to  speak  more  generally,  there  is  no  war- 
rant for  the  extremely  subjective  tendency  of  much  modem  spe- 
culation. Whenever,  in  the  view  of  the  intellect,  different 
hypotheses  are  equally  consistent  with  an  observed  fact,  the 
instinctive  testimony  of  consciousness  as  to  their  relative  value 
must  be  allowed  to  possess  authority. 

3rdly.  If  the  study  of  the  laws  of  thought  avails  us  neither 
to  determine  the  actual  constitution  of  things,  nor  to  explain  the 
facts  involved  in  that  constitution  which  have  perplexed  the  wise 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT.  419 

and  saddened  the  thoughtful  in  all  ages, — still  less  does  it  enable 
us  to  rise  above  the  present  conditions  of  our  being,  or  lend  its 
sanction  to  the  doctrine  which  affirms  the  possibility  of  an  in- 
tuitive knowledge  of  the  infinite,  and  the  unconditioned, — whe- 
ther such  knoAvledge  be  sought  for  in  the  realm  of  Nature,  or 
above  that  realm.  We  can  never  be  said  to  comprehend  that 
which  is  represented  to  thought  as  the  limit  of  an  indefinite 
process  of  abstraction.  A progression  ad  infinitum  is  impos- 
sible to  finite  powers.  But  though  we  cannot  comprehend  the 
infinite,  there  may  be  even  scientific  grounds  for  believing  that 
human  nature  is  constituted  in  some  relation  to  the  infinite.  We 
cannot  perfectly  express  the  laws  of  thought,  or  establish  in  the 
most  general  sense  the  methods  of  which  they  form  the  basis,  with- 
out at  least  the  implication  of  elements  which  ordinary  language 
expresses  by  the  terms  “ Universe”  and  “ Eternity.”  As  in  the 
pure  abstractions  of  Geometry,  so  in  the  domain  of  Logic  it  is 
seen,  that  the  empire  of  Truth  is,  in  a certain  sense,  larger  than 
that  of  Imagination.  And  as  there  are  many  special  departments 
of  knowledge  which  can  only  be  completely  surveyed  from  an  ex- 
ternal point,  so  the  theory  of  the  intellectual  processes,  as  applied 
only  to  finite  objects,  seems  to  involve  the  recognition  of  a 
sphere  of  thought  from  which  all  limits  are  withdrawn.  If  then, 
on  the  one  hand,  we  cannot  discover  in  the  laws  of  thought  and 
their  analogies  a sufficient  basis  of  proof  for  the  conclusions  of 
a too  daring  mysticism  ; on  the  other  hand  we  should  err  in  re- 
garding them  as  wholly  unsuggestive.  As  parts  of  our  intellec- 
tual nature,  it  seems  not  improbable  that  they  should  manifest 
their  presence  otherwise  than  by  merely  prescribing  the  condi- 
tions of  formal  inference.  Whatever  grounds  we  have  for  con- 
necting them  with  the  peculiar  tendencies  of  physical  speculation 
among  the  Ionian  and  Italic  philosophers,  the  same  grounds 
exist  for  associating  them  with  a disposition  of  thought  at  once 
more  common  and  more  legitimate.  To  no  casual  influences,  at 
least,  ought  we  to  attribute  that  meditative  spirit  which  then 
most  delights  to  commune  with  the  external  magnificence  of 
Nature,  when  most  impressed  with  the  consciousness  of  sempi- 
ternal verities, — which  reads  in  the  nocturnal  heavens  a bright 
manifestation  of  order ; or  feels  in  some  wild  scene  among  the 


420  CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

hills,  the  intimations  of  more  than  that  abstract  eternity  which 
had  rolled  away  ere  yet  their  dark  foundations  were  laid.* 

9.  Refraining  from  the  further  prosecution  of  a train  of  thought 
which  to  some  may  appear  to  be  of  too  speculative  a character, 
let  us  briefly  review  the  positive  results  to  which  we  have  been  led. 
It  has  appeared  that  there  exist  in  our  nature  faculties  which 
enable  us  to  ascend  from  the  particular  facts  of  experience  to  the 
general  propositions  which  form  the  basis  of  Science  ; as  well  as 
faculties  whose  office  it  is  to  deduce  from  general  propositions 
accepted  as  true  the  particular  conclusions  which  they  involve. 
It  has  been  seen,  that  those  faculties  are  subject  in  their  opera- 
tions to  laws  capable  of  precise  scientific  expression,  but  invested 
with  an  authority  which,  as  contrasted  with  the  authority  of  the 
laws  of  nature,  is  distinct,  sui  generis , and  underived.  Further, 
there  has  appeared  to  be  a manifest  fitness  between  the  intel- 
lectual procedure  thus  made  known  to  us,  and  the  conditions  of 
that  system  of  things  by  which  we  are  surrounded, — such  condi- 
tions, I mean,  as  the  existence  of  species  connected  by  general 
resemblances,  of  facts  associated  under  general  laws ; together 
with  that  union  of  permanency  with  order,  which  while  it  gives 
stability  to  acquired  knowledge,  lays  a foundation  for  the  hope 
of  indefinite  progression.  Human  nature,  quite  independently 
of  its  observed  or  manifested  tendencies,  is  seen  to  be  constituted 
in  a certain  relation  to  Truth  ; and  this  relation,  considered  as  a 
subject  of  speculative  knowledge,  is  as  capable  of  being  studied 
in  its  details,  is,  moreover,  as  worthy  of  being  so  studied,  as  are 
the  several  departments  of  physical  science,  considered  in  the  same 
aspect.  I would  especially  direct  attention  to  that  view  of  the 
constitution  of  the  intellect  which  represents  it  as  subject  to  laws 
determinate  in  their  character,  but  not  operating  by  the  power  of 
necessity;  which  exhibits  it  as  redeemed  from  the  dominion  of 
fate,  without  being  abandoned  to  the  lawlessness  of  chance.  We 
cannot  embrace  this  view  without  accepting  at  least  as  probable 
the  intimations  which,  upon  the  principle  of  analogy,  it  seems  to 
furnish  respecting  another  and  a higher  aspect  of  our  nature, — its 
subjection  in  the  sphere  of  duty  as  well  as  in  that  of  knowledge  to 


* Psalm  xc.  2. 


CHAP.  XXII.]  CONSTITUTION  OF  THE  INTELLECT.  421 

fixed  laws  whose  authority  does  not  consist  in  power, — its  con- 
stitution with  reference  to  an  ideal  standard  and  a final  purpose. 
It  has  been  thought,  indeed,  that  scientific  pursuits  foster  a dis- 
position either  to  overlook  the  specific  differences  between  the 
moral  and  the  material  world,  or  to  regard  the  former  as  in  no  pro- 
per sense  a subject  for  exact  knowledge.  Doubtless  all  exclusive 
pursuits  tend  to  produce  partial  views,  and  it  may  be,  that  a mind 
long  and  deeply  immersed  in  the  contemplation  of  scenes  over 
which  the  dominion  of  a physical  necessity  is  unquestioned  and  su- 
preme, may  admit  with  difficulty  the  possibility  of  another  order  of 
things.  But  it  is  because  of  the  exclusiveness  of  this  devotion  to  a 
particular  sphere  of  knowledge,  that  the  prejudice  in  question 
takes  possession,  if  at  all,  of  the  mind.  The  application  of 
scientific  methods  to  the  study  of  the  intellectual  phenomena, 
conducted  in  an  impartial  spirit  of  inquiry,  and  without  over- 
looking those  elements  of  error  and  disturbance  which  must  be 
accepted  as  facts , though  they  cannot  be  regarded  as  laws , in 
the  constitution  of  our  nature,  seems  to  furnish  the  materials  of 
a juster  analogy. 

10.  If  it  be  asked  to  what  practical  end  such  inquiries  as  the 
above  point,  it  may  be  replied,  that  there  exist  various  objects, 
in  relation  to  which  the  courses  of  men’s  actions  are  mainly  de- 
termined by  their  speculative  views  of  human  nature.  Educa- 
tion, considered  in  its  largest  sense,  is  one  of  those  objects.  The 
ultimate  ground  of  all  inquiry  into  its  nature  and  its  methods 
must  be  laid  in  some  previous  theory  of  what  man  is,  what  are 
the  ends  for  which  his  several  faculties  were  designed,  what 
are  the  motives  which  have  power  to  influence  them  to  sustained 
action,  and  to  elicit  their  most  perfect  and  most  stable  results. 
It  may  be  doubted,  whether  these  questions  have  ever  been 
considered  fully,  and  at  the  same  time  impartially,  in  the  rela- 
tions here  suggested.  The  highest  cultivation  of  taste  by  the 
study  of  the  pure  models  of  antiquity,  the  largest  acquaintance 
with  the  facts  and  theories  of  modern  physical  science,  viewed 
from  this  larger  aspect  of  our  nature,  can  only  appear  as  parts  of 
a perfect  intellectual  discipline.  Looking  from  the  same  point 
of  view  upon  the  means  to  be  employed,  we  might  be  led  to  in- 
quire, whether  that  all  but  exclusive  appeal  which  is  made  in 


422 


CONSTITUTION  OF  THE  INTELLECT.  [CHAP.XXIf. 

the  present  day  to  the  spirit  of  emulation  or  cupidity,  does  not 
tend  to  weaken  the  influence  of  those  more  enduring  motives 
which  seem  to  have  been  implanted  in  our  nature  for  the  imme- 
diate end  in  view.  Upon  these,  and  upon  many  other  questions, 
the  just  limits  of  authority,  the  reconciliation  of  freedom  of 
thought  with  discipline  of  feelings,  habits,  manners,  and  upon 
the  whole  moral  aspect  of  the  question, — what  unfixedness  of 
opinion,  what  diversity  of  practice,  do  we  meet  with!  Yet,  in 
the  sober  view  of  reason,  there  is  no  object  within  the  compass 
of  human  endeavours  which  is  of  more  weight  and  moment  than 
this,  considered,  as  I have  said,  in  its  largest  meaning.  Now, 
whatsoever  tends  to  make  more  exact  and  definite  our  view  of 
human  nature,  in  any  of  its  real  aspects,  tends,  in  the  same  pro- 
portion, to  reduce  these  questions  into  narrower  compass,  and 
restrict  the  limits  of  their  possible  solution.  Thus  may  even 
speculative  inquiries  prove  fruitful  of  the  most  important  prin- 
ciples of  action. 

11.  Perhaps  the  most  obviously  legitimate  bearing  of  such 
speculations  would  be  upon  the  question  of  the  place  of  Mathe- 
matics in  the  system  of  human  knowledge,  and  the  nature 
and  office  of  mathematical  studies,  as  a means  of  intellectual 
discipline.  No  one  who  has  attended  to  the  course  of  recent 
discussions  can  think  this  question  an  unimportant  one.  Those 
who  have  maintained  that  the  position  of  Mathematics  is  in 
both  respects  a fundamental  one,  have  drawn  one  of  their  strongest 
arguments  from  the  actual  constitution  of  things.  The  mate- 
rial frame  is  subject  in  all  its  parts  to  the  relations  of  number. 
All  dynamical,  chemical,  electrical,  thermal,  actions,  seem  not 
only  to  be  measurable  in  themselves,  but  to  be  connected  with 
each  other,  even  to  the  extent  of  mutual  convertibility,  by  nu- 
merical relations  of  a perfectly  definite  kind.  But  the  opinion 
in  question  seems  to  me  to  rest  upon  a deeper  basis  than  this. 
The  laws  of  thought,  in  all  its  processes  of  conception  and  of 
reasoning,  in  all  those  operations  of  which  language  is  the  ex- 
pression or  the  instrument,  are  of  the  same  kind  as  are  the  laws 
of  the  acknowledged  processes  of  Mathematics.  It  is  not  con- 
tended that  it  is  necessary  for  us  to  acquaint  ourselves  with  those 
laws  in  order  to  think  coherently,  or,  in  the  ordinary  sense  of 


CHAP  XXIX.]  CONSTITUTION  OF  THE  INTELLECT.  423 

the  terms,  to  reason  well.  Men  draw  inferences  without  any 
consciousness  of  those  elements  upon  which  the  entire  procedure 
depends.  Still  less  is  it  desired  to  exalt  the  reasoning  faculty 
over  the  faculties  of  observation,  of  reflection,  and  of  judgment. 
But  upon  the  very  ground  that  human  thought,  traced  to  its 
ultimate  elements,  reveals  itself  in  mathematical  forms,  we  have 
a presumption  that  the  mathematical  sciences  occupy,  by  the 
constitution  of  our  nature,  a fundamental  place  in  human  know- 
ledge, and"  that  no  system  of  mental  culture  can  be  complete  or 
fundamental,  which  altogether  neglects  them. 

But  the  very  same  class  of  considerations  shows  with  equal 
force  the  error  of  those  who  regard  the  study  of  Mathematics, 
and  of  their  applications,  as  a sufficient  basis  either  of  knowledge 
or  of  discipline.  If  the  constitution  of  the  material  frame  is  ma- 
thematical, it  is  not  merely  so.  If  the  mind,  in  its  capacity  of 
formal  reasoning,  obeys,  whether  consciously  or  unconsciously, 
mathematical  laws,  it  claims  through  its  other  capacities  of  sen- 
timent and  action,  through  its  perceptions  of  beauty  and  of 
moral  fitness,  through  its  deep  springs  of  emotion  and  affection, 
to  hold  relation  to  a different  order  of  things.  There  is,  more- 
over, a breadth  of  intellectual  vision,  a power  of  sympathy  with 
truth  in  all  its  forms  and  manifestations,  which  is  not  measured 
by  the  force  and  subtlety  of  the  dialectic  faculty.  Even  the 
revelation  of  the  material  universe  in  its  boundless  magnitude, 
and  pervading  order,  and  constancy  of  law,  is  not  necessarily  the 
most  fully  apprehended  by  him  who  has  traced  with  minutest 
accuracy  the  steps  of  the  great  demonstration.  And  if  we  em- 
brace in  our  survey  the  interests  and  duties  of  life,  how  little  do 
any  processes  of  mere  ratiocination  enable  us  to  comprehend  the 
weightier  questions  which  they  present ! As  truly,  therefore,  as 
the  cultivation  of  the  mathematical  or  deductive  faculty  is  a part 
of  intellectual  discipline,  so  truly  is  it  only  a part.  The  pre- 
judice which  would  either  banish  or  make  supreme  any  one 
department  of  knowledge  or  faculty  of  mind,  betrays  not  only 
error  of  judgment,  but  a defect  of  that  intellectual  modesty 
which  is  inseparable  from  a pure  devotion  to  truth.  It  assumes 
the  office  of  criticising  a constitution  of  things  which  no  human 
appointment  has  established,  or  can  annul.  It  sets  aside  the 


424  CONSTITUTION  OF  THE  INTELLECT.  [CHAP.  XXII. 

ancient  and  just  conception  of  truth  as  one  though  manifold. 
Much  of  this  error,  as  actually  existent  among  us,  seems  due 
to  the  special  and  isolated  character  of  scientific  teaching — 
which  character  it,  in  its  turn,  tends  to  foster.  The  study  of 
philosophy,  notwithstanding  a few  marked  instances  of  exception, 
has  failed  to  keep  pace  with  the  advance  of  the  several  depart- 
ments of  knowledge,  whose  mutual  relations  it  is  its  province  to 
determine.  It  is  impossible,  however,  not  to  contemplate  the 
particular  evil  in  question  as  part  of  a larger  system,  and  connect 
it  with  the  too  prevalent  view  of  knowledge  as  a merely  secular 
thing,  and  with  the  undue  predominance,  already  adverted  to,  of 
those  motives,  legitimate  within  their  proper  limits,  which  are 
founded  upon  a regard  to  its  secular  advantages.  In  the  extreme 
case  it  is  not  difficult  to  see  that  the  continued  operation  of 
such  motives,  uncontrolled  by  any  higher  principles  of  action, 
uncorrected  by  the  personal  influence  of  superior  minds,  must 
tend  to  lower  the  standard  of  thought  in  reference  to  the  objects 
of  knowledge,  and  to  render  void  and  ineffectual  whatsoever  ele- 
ments of  a noble  faith  may  still  survive.  And  ever  in  proportion 
as  these  conditions  are  realized  must  the  same  effects  follow. 
Hence,  perhaps,  it  is  that  we  sometimes  find  juster  conceptions 
of  the  unity,  the  vital  connexion,  and  the  subordination  to  a 
moral  purpose,  of  the  different  parts  of  Truth,  among  those  who 
acknowledge  nothing  higher  than  the  changing  aspect  of  col- 
lective humanity,  than  among  those  who  profess  an  intellectual 
allegiance  to  the  Father  of  Lights.  But  these  are  questions 
which  cannot  further  be  pursued  here.  To  some  they  will  ap- 
pear foreign  to  the  professed  design  of  this  work.  But  the 
consideration  of  them  has  arisen  naturally,  either  out  of  the 
speculations  which  that  design  involved,  or  in  the  course  of 
reading  and  reflection  which  seemed  necessary  to  its  accomplish- 
ment. 


THE  END. 


':J  . 


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